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b/39E2T4oBgHgl3EQfOAbJ/content/tmp_files/2301.03744v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..86f7108d83fc1baceafa8155c2d9c3d1a9c75477 --- /dev/null +++ b/39E2T4oBgHgl3EQfOAbJ/content/tmp_files/2301.03744v1.pdf.txt @@ -0,0 +1,1562 @@ +Inflation in Weyl Scaling Invariant Gravity with R3 Extensions +Qing-Yang Wanga, Yong Tanga,b,c,d, and Yue-Liang Wua,b,c,e +aUniversity of Chinese Academy of Sciences (UCAS), Beijing 100049, China +bSchool of Fundamental Physics and Mathematical Sciences, +Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China +cInternational Center for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China +dNational Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China +eInstitute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China +(Dated: January 11, 2023) +Abstract +The cosmological observations of cosmic microwave background and large-scale structure indicate +that our universe has a nearly scaling invariant power spectrum of the primordial perturbation. +However, the exact origin for this primordial spectrum is still unclear. +Here, we propose the +Weyl scaling invariant R2 + R3 gravity that gives rise to inflation that is responsible for the +primordial perturbation in the early universe. We develop both analytic and numerical treatments +on inflationary observables, and find this model gives a distinctive scalar potential that can support +two different patterns of inflation. The first one is similar to that occurs in the pure R2 model, +but with a wide range of tensor-to-scalar ratio r from O(10−4) to O(10−2). The other one is a new +situation with not only slow-roll inflation but also a short stage of oscillation-induced accelerating +expansion. Both patterns of inflation have viable parameter spaces that can be probed by future +experiments on cosmic microwave background and primordial gravitational waves. +1 +arXiv:2301.03744v1 [astro-ph.CO] 10 Jan 2023 + +I. +INTRODUCTION +Inflation is a hypothetical epoch of exponential expansion introduced in the very early +universe to solve the cosmological horizon and flatness problems [1, 2]. It is also a reasonable +scheme to explain the origin of primordial density perturbations, which plays the role of +the seeds that formed the structure of current universe [3]. In recent years, the precise +measurement of cosmic microwave background (CMB) presents us with an almost scale +invariant spectrum of primordial perturbations [4]. This result is usually explained by an +approximate de Sitter spacetime of the very early universe [5–9]. Moreover, it is theoretically +explored that there is a more profound and basic principle behind the phenomenon, namely, +local Weyl scaling invariance of the universe. This symmetry is first proposed by H. Weyl in +the attempt of understanding gravity and electromagnetism in a unified framework [10, 11], +and after a century of development, it has been applied extensively to particle physics, +cosmology [12–30] and gauge theory of gravity [31–34]. +Lately, inflation in the Weyl scaling invariant theory of gravity, especially induced by +a quadratic curvature term R2, has been of many concern [35–45]. Comparing with the +conventional R2 model, which is also called Starobinsky model [46–49], the scaling invariant +version not only allows a viable inflation scenario with good observational agreement, but +also provides a framework to comprehend another fundamental puzzles, such as hierarchy +problem [37, 40, 50] and dark matter candidates [41, 45]. +However, inflation with only quadratic scalar curvature might be just a simplistic scenario. +From the viewpoint of effective field theory, any higher-order curvature effects may exist and +play a role in the early universe. Hence it is reasonable to evaluate their impacts on inflation. +Generally, the extensions with high-order tensors, like RµνRµν or RµνρσRµνρσ, can result in +unacceptable ghost degrees of freedom [51], while the terms of arbitrary functions of the +Ricci scalar are known to be safe. Therefore, in this paper, we consider a minimal extension +of Ricci scalar beyond the R2 model with Weyl scaling invariance, namely a cubic term +coupled with an extra scalar field as denominator R3/ϕ2. We will show that even if this +term is extremely small, it will have an essential impact on inflation, which even open up a +completely different inflationary scenario from Weyl R2 and conventional R2 + R3 models. +The paper is organized as follows. In Sec. II, we develop the analytic formalism of Weyl +R2 + R3 model and derive the effective scalar potential. We show that in some cases, the +2 + +potential has two different kinds of global minima, leading to two distinctive inflationary pat- +terns. In Sec. III, we investigate the inflation in the pattern of evolving to the side minimum. +We calculate the spectral index ns and tensor-to-scalar ratio r of the inflationary perturba- +tions, and give the preferred parameter space allowed by the latest observations. Analytical +treatments are developed for more transparent, physical understanding of the asymptotic +behaviors. Then in Sec. IV, we investigate the pattern of evolving to the center minimum. +A special process called “oscillating inflation” is considered in detail. Finally, conclusions +are given in Sec. V. We adopt the following conventions: metric ηµν = (−1, +1, +1, +1), +natural unit ℏ = c = 1 and MP ≡ 1/ +√ +8πG = 2.435 × 1018 GeV = 1. +II. +WEYL SCALING INVARIANT R2 + R3 MODEL +We start with the following Lagrangian for metric field gµν, scalar field ϕ, and Weyl gauge +field Wµ ≡ gWwµ with local scaling symmetry +L +√−g = 1 +2 +� +ϕ2 ˆR + α ˆR2 + β +ϕ2 ˆR3 +� +− ζ +2DµϕDµϕ − +1 +4g2 +W +FµνF µν. +(1) +Here g is the determinant of metric, α, β and ζ are constant parameters, Dµ = ∂µ − Wµ +is the covariant derivative associated with scaling symmetry, gW is the coupling constant, +Fµν ≡ ∂µWν − ∂νWµ defines the invariant field strength of Wµ, and ˆR is the Ricci scalar +defined by the local scaling invariant connection +ˆΓρ +µν = 1 +2gρσ [(∂µ + 2Wµ)gσν + (∂ν + 2Wν)gµσ − (∂σ + 2Wσ)gµν] . +(2) +Explicit calculation shows the relation between ˆR and usual R defined by metric field gµν, +ˆR = R − 6WµW µ − +6 +√−g∂µ(√−gW µ). +(3) +It is straightforward to verify the invariance of Eq. (1) under the following Weyl scaling +transformation +metric : gµν → g′ +µν = f 2(x)gµν, +scalar : φ → φ′ = f −1(x)φ, +Ricci scalar : +ˆR → ˆR′ = f −2(x) ˆR, +Weyl vector : Wµ → W ′ +µ = Wµ − ∂µ ln f(x), +(4) +where f(x) is an arbitrary positive function. +3 + +The purpose to explore the Lagrangian in Eq. (1) is two-fold. +Theoretically, such a +ˆR3 term constitutes as a simple extension of the ˆR2 theory, motivated from perspective of +effective field theories and also quantum loop corrections in more fundamental theories [31– +34]. Phenomenologically, it is worthwhile to explore how such a term would modify the +cosmological observations related to inflation, and evaluate the likelihood and robustness of +the predictions in the lowest-order theories. +A. +Formalism in Einstein frame +General f(R) gravity is equivalent to the Einstein gravity with a scalar field [52, 53]. In +Ref. [41], we have extended the proof in general scaling invariant F( ˆR, ϕ) gravity. We can +explicitly show that by introducing an auxiliary scalar field χ and rewrite the high-order +curvature terms as +F( ˆR, ϕ) ≡ ϕ2 ˆR + α ˆR2 + β +ϕ2 ˆR3 = F ˆR( ˆR → χ2, ϕ)( ˆR − χ2) + F( ˆR → χ2, ϕ). +(5) +Here F ˆR denotes the derivative over ˆR, F ˆR = ∂F( ˆR, ϕ)/∂ ˆR. We can verify that the equiv- +alence relation χ2 = ˆR can be obtained from the Euler-Lagrange equation, δL +δχ = 0. Substi- +tuting Eq. (5) into Eq. (1), we find +L +√−g = 1 +2 +� +ϕ2 + 2αχ2 + 3β +ϕ2 χ4 +� +ˆR − 1 +2 +� +αχ4 + 2β +ϕ2 χ6 +� +− ζ +2DµϕDµϕ − +1 +4g2 +W +FµνF µν. +(6) +Now we have demonstrated that linearization of ˆR has led to the non-minimal coupling of +the scalar field, χ. +We can transform the above Lagrangian into the Einstein frame by making a Weyl or +conformal transformation of the metric field. However, we note that scaling invariance is +still preserved in our model with χ → χ′ = f −1(x)χ. Therefore, we can directly normalize +the coefficient before the Ricci scalar as +ϕ2 + 2αχ2 + 3βχ4/ϕ2 = 1, +(7) +due to the scaling invariance of Eq. (6). This is equivalent to making a Weyl transformation +with f(x) = +� +ϕ2 + 2αχ2 + 3βχ4/ϕ2 in Eq. (4). Further dropping the total derivative term +4 + +in Eq. (3) due to its null surface integral, we can write the Lagrangian as +L +√−g =1 +2R − ζ +2DµϕDµϕ − V (ϕ) − +1 +4g2 +W +FµνF µν − 3W µWµ +=R +2 − +∂µϕ∂µϕ +2/ζ + ϕ2/3 − V (ϕ) − +1 +4g2 +W +FµνF µν − 6 + ζϕ2 +2 +� +Wµ − ∂µ ln |6 + ζϕ2| +2 +�2 +, +(8) +with the scalar potential +V (ϕ) = α +2 χ4 + β +ϕ2χ6 = α +6β +� +ϕ4 − ϕ2� ++ α3ϕ4 +27β2 +�� +1 − 3β +α2 +� +1 − ϕ−2��3/2 +− 1 +� +, +(9) +where we have solved χ from Eq. (7) +χ2 = αϕ2 +3β +�� +1 − 3β +α2 (1 − ϕ−2) − 1 +� +. +(10) +It is now clear that we have a minimally-coupled scalar ϕ with a non-canonical kinetic +term. To further simplifying the theoretical formalism, we introduce the following redefini- +tions for the scalar and the Weyl gauge field +ϕ2 ≡ +� +� +� +� +� +6 +|ζ| sinh2 � +±Φ +√ +6 +� +for ζ > 0, +6 +|ζ| cosh2 � +±Φ +√ +6 +� +for ζ < 0, +(11) +˜Wµ ≡ Wµ − 1 +2∂µ ln |6 + ζϕ2| ≡ gW ˜wµ. +(12) +Then the final Lagrangian turns into a more compact form +L +√−g = 1 +2R − 1 +2∂µΦ∂µΦ − V (Φ) − +1 +4g2 +W +˜Fµν ˜F µν − 1 +2m2(Φ) ˜W µ ˜Wµ, +(13) +with the mass term of Weyl gauge field +m2(Φ) = +� +� +� +� +� ++6 cosh2 � +Φ +√ +6 +� +for ζ > 0, +−6 sinh2 � +Φ +√ +6 +� +for ζ < 0. +(14) +We should note that m2(Φ) is negative when ζ < 0. Therefore, to avoid Weyl gauge boson +becoming tachyonic in this case, it requires some other mechanisms to obtain a real mass, +for example, introducing other scalar field, which we do not explore in this paper. For viable +inflation, both positive and negative are possible, as we shall show later. +In the above discussion, we have demonstrated that Weyl scaling invariant ˆR2+ ˆR3 model +can be written equivalently as the Einstein gravity coupled with a self-interacting scalar Φ +5 + +and a massive vector ˜Wµ with a field-dependent mass. This conclusion is also true for any +Weyl scaling invariant model of gravity with high-order curvature ˆRn as the above formalism +applies straightforwardly. It is also worth to point out that Weyl vector boson can serve +as a dark matter candidate [27, 28, 41], with details of the relic abundance being discussed +in [45]. In this paper, we shall concentrate on the scalar potential Eq. (9) and discuss the +viable inflation scenarios with the presence of ˆR3. +B. +Effective scalar potentials +There are two necessary requirements for the potential Eq. (9). The first one is ϕ2 > 0 +since ϕ is a real scalar field. The other is 1 − 3β +α2 +� +1 − +1 +ϕ2 +� +≥ 0, otherwise an imaginary +potential will emerge. Consequently, there are some constraints on the parameters and the +viable value of Φ. We can rewrite the second requirement as +sinh +�±Φ +√ +6 +� +≥ or ≤ +� +|ζ| +6 − 2α2/β , for ζ > 0, +cosh +�±Φ +√ +6 +� +≥ or ≤ +� +|ζ| +6 − 2α2/β , for ζ < 0, +(15) +where “ ≥ ” for β < α2 +3 and “ ≤ ” for β ≥ α2 +3 . For convenience, we define λ ≡ +� +|ζ| +6−2α2/β +and γ ≡ +3β +α2, then discuss the possible ranges of the potential corresponding to different +parameters. The results are listed in the Table. I. To ensure the theoretical stability, we +require that Φ can only evolve within these ranges where the potential is real. Fig. 1 shows +some instances of the scalar potential for several values of ζ and γ. +We first discuss the case of positive ζ. When γ = 0, it is a hill-top-like potential with two +minima at Φ = ± +√ +6 sinh−1 � +ζ +6. However, as long as there is a tiny cubic curvature, whether +positive or negative, the shape of potential will be affected significantly. When γ > 0, the +potential turns to decrease near Φ = 0, and a third vacuum can form there. This behavior +is transparent, because when ζ > 0, Φ = 0 corresponds to ϕ2 = 0 according to Eq. (11), +then substituting it in Eq. (9) will obtain V |Φ=0 = 0. When γ < 0, the potential turns to +rise near Φ = 0 and become imaginary and unphysical in − +√ +6 sinh−1 λ < Φ < +√ +6 sinh−1 λ, +which has been listed in Table. I. +Next, we switch to the case of negative ζ. It is evident in Fig. 1 that when ζ < 0 and |ζ| +or |γ| is relatively small, the modification of ˆR3 term on the Weyl R2 potential is moderate, +6 + +TABLE I. Effective potential range of the Weyl R2 + R3 model. +ζ +γ or β +real V (ϕ) +ζ > 0 +γ ≥ 1 +|Φ| ≤ +√ +6 sinh−1 λ +0 ≤ γ < 1 +fully real +γ < 0 +|Φ| ≥ +√ +6 sinh−1 λ +−6 < ζ < 0 +γ > +1 +1+ζ/6 +fully imaginary +1 < γ ≤ +1 +1+ζ/6 +|Φ| ≤ +√ +6| cosh−1 λ| +γ ≤ 1 +fully real +ζ ≤ −6 +γ ≥ 1 +|Φ| ≤ +√ +6| cosh−1 λ| +1 +1+ζ/6 < γ < 1 +fully real +γ ≤ +1 +1+ζ/6 +|Φ| ≥ +√ +6| cosh−1 λ| +unlike the dramatic change near Φ = 0 in the case of positive ζ. This is because the mapping +of Φ ⇒ ϕ2 does not cover the interval of ϕ2 < 1 for ζ < 0 according to Eq. (11). In other +words, for negative ζ with modest |γ|, Φ → 0 does not lead to ϕ2 → 0, which brings the +violent behavior of the potential around here in the case of ζ > 0. However, when ζ is +excessively negative or |γ| is large enough, the violent variation will reappear to a certain +extent. For γ > 0, the potential will return to a downward trend near Φ = 0, albeit there +is no true vacuum formed (but a false vacuum is formed). And for excessively negative γ, +the imaginary potential will reappear in the range of − +√ +6| cosh−1 λ| < Φ < +√ +6| cosh−1 λ|, +which we have listed this situation in Table. I (see ζ ≤ −6 with γ ≤ +1 +1+ζ/6 case). +Generally, inflation takes place when the potential is flat and Φ evolves to the vacuum +(Φ|V =0). The cosmological observations would restrict the potential and the initial value Φi +when inflation starts, here the Φi is defined as the value when the comoving horizon of the +inflationary universe shrinks to the same size as today. +For ζ > 0 and γ > 0, the scalar potential contains three separate vacua, one lying at the +center and the other two at both sides. Therefore, there are two different viable inflationary +patterns. One pattern refers to the evolution into the central minimum, and the other into +the side minima. We can calculate the value of Φ which corresponds to the hill-top of the +7 + +-10 +-5 +0 +5 +10 +0 +0.5 +1 +1.5 +2 +10-10 +-10 +-5 +0 +5 +10 +0 +0.5 +1 +1.5 +2 +10-10 +-10 +-5 +0 +5 +10 +0 +0.5 +1 +1.5 +2 +10-10 + +-10 +-5 +0 +5 +10 +0 +0.5 +1 +1.5 +2 +10-10 +FIG. 1. Effective potentials of Weyl R2 + R3 model with α = 109 and various γ and ζ. Here we +only depict the real ranges of potentials. +potential in this case +Φh = ± +√ +6 sinh−1 +� +ζ +12 +√3γ − 2γ +3 − 4γ +, +(16) +which is the critical point of two inflationary patterns. Neglecting the velocity, if the initial +value of inflation field satisfies |Φi| > |Φh|, it will evolve towards the side vacua. If |Φi| < |Φh| +at the beginning, the inflation field will evolve towards the central vacuum. +For other cases of ζ and γ, there are only the global side minima. Hence the only feasible +inflationary pattern is that Φ evolves to either one of the side minimum. The initial value +Φi has to correspond to a real potential, and when there is a false vacuum in ζ < 0 case, it +requires a large enough |Φi| outside two local maxima of the potential to ensure the gradient +of V (Φi) towards the true vacuum. Next, we are going to discuss the inflation in these two +patterns respectively. +8 + +III. +INFLATION TO THE SIDE +In this inflation pattern, ϕ2 (defined as Eq. (11)) is usually not very close to 0, and as +we shall show later, observations generally would require an extremely small cubic curva- +ture, namely |γ| ≪ 1. Therefore in many cases, |γ(1 − ϕ−2)| ≪ 1 is satisfied. Under this +condition, we are able to have analytical treatment and expand the potential Eq. (9) as +V (ϕ) =ϕ4 − ϕ2 +2αγ ++ +ϕ4 +3αγ2 +� +−3γ +2 +� +1 − 1 +ϕ2 +� ++ 3γ2 +8 +� +1 − 1 +ϕ2 +�2 ++ γ3 +16 +� +1 − 1 +ϕ2 +�3 ++ O +�γ4 +ϕ8 +�� += 1 +8α +� +1 − ϕ2�2 +� +1 + γ +6 +� +1 − 1 +ϕ2 +� ++ O +�γ2 +ϕ4 +�� +. +(17) +Then with Eq. (11), we derive +V (Φ) = +� +� +� +� +� +1 +8α +� +1 − 6 +|ζ| sinh2 � +Φ +√ +6 +��2 � +1 + γ +6 +� +1 − |ζ| +6 csch2 � +Φ +√ +6 +�� ++ O(γ2) +� +for ζ > 0, +1 +8α +� +1 − 6 +|ζ| cosh2 � +Φ +√ +6 +��2 � +1 + γ +6 +� +1 − |ζ| +6 sech2 � +Φ +√ +6 +�� ++ O(γ2) +� +for ζ < 0. +(18) +The first term is exactly the effective potential of Weyl ˆR2, which has been shown in [41, 45], +and the rest originates from the cubic curvature term ˆR3, to the leading order of γ. Next +we shall calculate the inflationary physical quantities, the spectral index ns and tensor-to- +scalar ratio r, and contrast them with the latest observations. We first give an analytical +calculation for two limiting cases, then show the full numerical results for general cases. +A. +Analytical approach of γ → 0 case +We first discuss the γ → 0 case and show how ζ affects ns and r. The slow-roll parameters +in this case can be derived as +ϵ ≡ 1 +2 +�V ′(Φ) +V +�2 += +12 sinh2 � +2Φ +√ +6 +� +� +|ζ + 3| − 3 − 6 sinh2 � +Φ +√ +6 +��2, +(19) +η ≡ V ′′(Φ) +V += +12 cosh +� +4Φ +√ +6 +� +− 4|ζ + 3| cosh +� +2Φ +√ +6 +� +� +|ζ + 3| − 3 − 6 sinh2 � +Φ +√ +6 +��2 +. +(20) +Generally, the slow-roll inflation occurs when ϵ and |η| is small enough, and it will end when +any of them evolves to ∼ 1. For the situation we are concerned with, ϵ breaks the slow-roll +9 + +limit before the other. Thus we derive the value of Φ when inflation ends according to ϵ = 1 +Φe = +� +3 +2 ln +� +2 +� +|ζ + 3|2 + 3 +√ +3 +− |ζ + 3| + +� +7 +3|ζ + 3|2 − 4|ζ + 3| +√ +3 +� +|ζ + 3|2 + 3 + 3 +� +. (21) +When |ζ| > O(102), which is a preferred range by the observational constraints as we will +show shortly, the above equation can be approximated as +Φe ≃ +� +3 +2 ln +� 1 +√ +3 +� +2 + +� +7 − 4 +√ +3 − +√ +3 +� +|ζ + 3| +� +≃ +� +3 +2 ln (0.3094|ζ + 3|) . +(22) +It is now clear that when |ζ| is large enough, Φe will be almost independent of the sign of ζ. +Next, we calculate initial value Φi, which is defined when the size of comoving horizon +during inflation shrinks to the present size. We first focus on the e-folding number of the +slow-roll inflation +N ≡ ln ae +ai +≃ +� Φe +Φi +dΦ +√ +2ϵ, +(23) +where ai/e ≡ a(Φi/e) is the cosmic scale factor when inflation starts/ends. +Substituting +Eq. (19) into it, we find +N = +(|ζ + 3| − 3) ln +� +tanh +� +Φ +√ +6 +�� +− 6 ln +� +cosh +� +Φ +√ +6 +�� +4 +����� +Φe +Φi += |ζ + 3| − 3 +4 +ln +� +�tanh +� 1 +2 ln(0.3094|ζ + 3|) +� +tanh +� +Φi +√ +6 +� +� +� − 3 +2 ln +� +�cosh +� 1 +2 ln(0.3094|ζ + 3|) +� +cosh +� +Φi +√ +6 +� +� +� . +(24) +For the circumstances we are concerned with, namely N ∼ (50, 60) and |ζ| > O(102), the +second term of Eq. (24) is much smaller than the first term, and it can be estimated as +∼ −2.3. Thus we derive +Φi ≃ +√ +6 tanh−1 +�� +1 − +2 +0.3094|ζ + 3| + 1 +� +e +−4(N+2.3) +|ζ+3|−3 +� +≡ +√ +6 tanh−1 Ω(ζ, N). +(25) +Here we have defined Ω(ζ, N) for later convenience. +When |ζ| ≫ 4N, it can be further approximated as Φi ≃ +� +3 +2 ln +|ζ| +2N+7.8. Substituting +Eq. (25) into Eq. (19) and (20), we find +ϵi = +48Ω2 +[(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2, +(26) +ηi =4 [(Ω4 − 1)|ζ + 3| + 3(Ω4 + 6Ω2 + 1)] +[(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2 +. +(27) +10 + +As a result, the tensor-to-scalar ratio r and spectral index ns of inflationary perturbations +in the γ → 0 limit are finally calculated as +r = 16ϵi = +768Ω2 +[(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2, +(28) +ns = 1 − 6ϵi + 2ηi = 1 + 8(Ω4 − 1)|ζ + 3| + 24(Ω4 − 6Ω2 + 1) +[(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2 +. +(29) +For N ∼ (50, 60) and |ζ| > O(102), We can approximate the expressions as +r ≃ r∗ − 54 +ζ2 , +(30) +ns ≃ n∗ +s − 11N +ζ2 , +(31) +where +r∗ ≃ +12 +(N + 3.55)2, n∗ +s ≃ 1 − +2 +N + 3.55 − +3 +(N + 3.55)2 +(32) +are the predictions of Starobinsky model (see Appendix A for an analytical derivation.). +Thus it is evident that the predictions of inflationary perturbations in our model will converge +to that of Starobinsky model when γ → 0 and ζ → ∞. As |ζ| decreases, the value of r and +ns will also decrease. We show this trend as the pink area in Fig. 2. According to the latest +observation [54], the lower limit of ns has been constrained to ∼ 0.959, hence it requires +|ζ| > 270 in this γ → 0 case. +B. +Analytical approach of ζ → ∞ case +Now we discuss the ζ → ∞ case and show how γ affects r and ns. When ζ is large enough, +the potential is greatly widened. The side vacua are far away from 0 and so are Φi and Φe +(e.g., Φi ∼ 5.4MP, Φe ∼ 9.8MP for ζ = 104). Therefore Eq. (11) can be approximated as +ϕ2 = 6 +|ζ| +� +eΦ/ +√ +6 ± e−Φ/ +√ +6 +2 +�2 +≃ e +√ +2/3 +� +Φ−√ +3/2 ln(2|ζ|/3) +� +≡ e +√ +2/3(Φ−Φ0). +(33) +Here and after, without losing generality, we may choose to evolve in the positive Φ region, +and denote Φ0 as the minimum in this region. Substituting it into Eq. (17), we have the +scalar potential for Φ ≫ 0 +V (Φ) = 1 +8α +� +1 − e +√ +2/3(Φ−Φ0)�2 � +1 + γ +6 +� +1 − e−√ +2/3(Φ−Φ0)� ++ O(γ2) +� +. +(34) +11 + +FIG. 2. The predictions of spectral index ns combined with tensor-to-scalar ratio r in the Weyl +R2 + R3 model with e-folding number N ∼ (50, 60). The pink area shows the results in the γ → 0 +case with various ζ. The yellow and green areas respectively show the ζ → ∞ and ζ = −650 cases +with various γ. The red line is the result with both γ → 0 and ζ → ∞, which is equivalent to the +Starobinsky model. The blue area is the latest observation constraint given by the BICEP/Keck +collaboration [54]. +Ignoring the O(γ2) terms, we give an approximate expression for the slow-roll parameters +ϵ ≡ 1 +2 +�V ′(Φ) +V +�2 +≃ +� +γe +√ +2/3(Φ−Φ0) − 2(γ + 6)e +√ +8/3(Φ−Φ0) + γ +�2 +3 +� +e +√ +2/3(Φ−Φ0) − 1 +�2 � +γ − (γ + 6)e +√ +2/3(Φ−Φ0)�2, +(35) +η ≡ V ′′(Φ) +V +≃ 6(γ + 4)e +√ +8/3(Φ−Φ0) − 8(γ + 6)e +√ +6(Φ−Φ0) + 2γ +3 +� +e +√ +2/3(Φ−Φ0) − 1 +�2 � +γ − (γ + 6)e +√ +2/3(Φ−Φ0)�. +(36) +In this case, the slow-roll inflation also ends at ϵ ∼ 1. To find the expression of Φe, we +further approximate Eq. (35) as +ϵ ≃ +e−√ +8/3(Φ−Φ0) � +γ − 12e +√ +8/3(Φ−Φ0)�2 +108 +� +e +√ +2/3(Φ−Φ0) − 1 +�2 +. +(37) +12 + +....... +→ 60= -650 +95% CL +0.03 +68% CL +5×1 +0.01 +500 +X + = 250 +0.003 +3x 10 +3 × 10 +0.001 +0.955 +0.96 +0.965 +0.97 +0.975 +ns0.980.1 +0← +N = 50 +8个VThen Φe can be derived as +Φe = Φ0 − +� +3 +2 ln +�√ +3 +γ +�� +2(2 + +√ +3)γ + 9 − 3 +�� +. +(38) +If γ is extremely small, we will find Φe ≃ Φ0 − 0.94MP. +Next, we derive the analytic formula for Φi in this case. The e-folding number of the +slow-roll inflation can be calculated with Eq. (37) as +N = − +�27 +4γ tanh−1 +�� γ +12e−√ 2 +3 (Φ−Φ0) +� +− 3 +8 ln +� +12 − γe−√ 8 +3 (Φ−Φ0)� +− +√ +6 +4 (Φ − Φ0) +���� +Φe +Φi +. (39) +Considering N ∼ (50, 60) and γ < O(10−3), the first term of the integral is dominant, while +the rest are the marginal terms which can be approximately treated as a constant, ∼ −2.7. +Hence we have +N ≃ +�27 +4γ +� +tanh−1 +�� γ +12e−√ +2/3(Φi−Φ0) +� +− tanh−1 +�� γ +12e−√ +2/3(Φe−Φ0) +�� +− 2.7, +(40) +and derive +Φi = Φ0 − +� +3 +2 ln +����� +�12 +γ tanh +� +tanh−1 +�� γ +12e−√ +2/3(Φe−Φ0) +� ++ +� +4γ +27(N + 2.7) +������ +≃ Φ0 − +� +3 +2 ln +����� +�12 +γ tanh +� +tanh−1 (0.622√γ) + +� +4γ +27(N + 2.7) +������ +≡ Φ0 − +� +3 +2 ln Θ(γ, N), +(41) +where we have defined Θ(γ, N) for later convenience. Then substituting it into Eq. (35) and +(36), we find +ϵi = [γΘ(1 + Θ) − 2(γ + 6)]2 +3 [1 − Θ]2 [γΘ − (γ + 6)]2, +(42) +ηi = 2γΘ3 + 6(γ + 4)Θ − 8(γ + 6) +3 [1 − Θ]2 [γΘ − (γ + 6)] +. +(43) +Finally, we derive r and ns of the inflationary perturbations in the ζ → ∞ limit +r = 16ϵi = 16 [γΘ(1 + Θ) − 2(γ + 6)]2 +3 [1 − Θ]2 [γΘ − (γ + 6)]2 , +(44) +ns = 1 − 6ϵi + 2ηi = 1 − 2 [γΘ(1 + Θ) − 2(γ + 6)]2 +[1 − Θ]2 [γΘ − (γ + 6)]2 + 4γΘ3 + 3(γ + 4)Θ − 4(γ + 6) +3 [1 − Θ]2 [γΘ − (γ + 6)] +. (45) +13 + +If γ is extremely small, smaller than O(10−4), the above expressions can be linearly approx- +imated as +r ≃ r∗ − 2.4γ, +(46) +ns ≃ n∗ +s − 0.42γN, +(47) +where r∗ and n∗ +s have been defined in the last paragraph of Sec. III.A. We can see that +compared with the predictions of Starobinsky model, a positive γ will reduce both r and +ns, while a negative γ will increase them. We show this trend as the yellow area in Fig. 2. +It is manifest that the observations have constrained |γ| ≲ 5 × 10−4 in this ζ → ∞ case. +Actually, this result agrees with other numerical investigations of the R3-extended Starobin- +sky model [55–61], since the potential Eq. (34) is the same as the R3-extended Starobinsky +model with a vacuum shift. Moreover, compared with Eq. (30) and (31), we note that the +predictions of r and ns in the γ → 0 case is similar to that of the ζ → ∞ and γ > 0 case +with a simple replacement of γ ↔ 24 +ζ2. This can be seen more clearly from Fig. 2, where the +pink area overlaps with the yellow area with γ > 0. +C. +General cases +Now we discuss the general cases with various ζ and γ by numerical treatment. The +results are shown in Fig. 3. Here the parameter ranges satisfying observational constraints +(see blue area in Fig. 2) are marked with colored areas, where the color gradient from blue to +red corresponds to ascending value of r. The gray areas represent that the potential defined +by these parameters cannot support an adequate inflation. In other words, their maximal +e-folding number is unable to reach N = 50 or 60. The white areas are the parameter ranges +that can give rise to ample inflation, but their prediction of ns or r has been excluded by +the observation constraints. Here we mark two dotted lines to distinguish the boundaries of +constraints. Beyond the pink one indicates a large ns that exceeds the observational upper +limit, while beyond the green one signifies a too small prediction. +Let us focus on the colored parameter ranges that are allowed by observations. In the +|ζ| ≫ 1000 case, the result is roughly equivalent to the analytical calculation shown in the +last subsection. The prediction of r is limited to 0.002 < r < 0.006. However, distinctive +situations appear when |ζ| is small. First, when −1000 < ζ < −200, the restrictions on +γ is relaxed, which can stand |γ| ∼ 6 × 10−3 at most. Besides, the upper limit of r is +14 + +FIG. 3. Possible parameter space for Weyl R2 + R3 model when Φ evolves to the side vacuum. +The colored areas are the parameter ranges allowed by the latest observations of BICEP/Keck +collaboration [54], where the color gradient from blue to red corresponds to r increases from 0.001 +to the observational upper limit 0.036. The dotted lines are the boundaries that ns exceeds the +observational upper (pink line) or lower (green line) limit. The gray areas represent the parameter +ranges with inadequate inflation, namely, the maximal e-folding number of inflation cannot reach +N = 50 or 60. +greatly expanded. There is even a small parameter range that gives r > 0.01. We show an +example as the green area in Fig. 2. It clearly shows a distinguishable feature from the Weyl +R2 model and the R3-extended Starobinsky model. If the next generation experiment of +CMB B-mode polarization detects the primordial gravitational waves with r > 0.01, it may +support Weyl R2 + R3 model. Another notable feature emerges at 0 < ζ < 200, where the +15 + +r0 +-2 +ns > 0.974 +-4 +inadequate e-folds +-6 +-3000 +-2000 +-1000 +0 +1000 +2000 +× 10-3 +4 +N = 60 +2 +ns < 0.959 +0 +-2 +ns > 0.974 +-4 +inadequate e-folds +9- +-3000 +-2000 +-1000 +0 +1000 +20000.000 +0.015 +3000 +0.005 +0.001 +3000×10-3 +4 +N = 50 +2 +ns < 0.959negative γ, even if very small, can greatly affect the predictions of primordial perturbations. +Actually, there are some cases with small positive ζ and small negative γ can give proper +r and ns that match the observation constraints, and generally, r is extremely small. For +instance, when ζ = 80, γ = −4 × 10−8, and N = 60, we have ns = 0.963 and r = 3 × 10−4. +IV. +INFLATION TO THE CENTER +As we mentioned earlier, the third vacuum appears at Φ = 0 in the case of ζ > 0 and +γ > 0, and if the initial value satisfies |Φi| < |Φh| (Φh is defined in Eq. (16)), inflation can +happen in the evolution of Φ to 0. Actually, the situation is more complicated. A process +called “oscillating inflation” [62–74] will continue immediately after the end of slow-roll +inflation because the scalar potential in this case is a non-convex function in the region close +to the vacuum, which means there is d2V +dΦ2 < 0 when Φ nears 0. In other words, for such a +non-convex potential, despite the slow-roll conditions (ϵ ≪ 1 and |η| ≪ 1) has been violated +during the bottom oscillation of the inflaton potential, the universe can keep accelerating +expansion until the average amplitude of the inflaton’s oscillation becomes lower than the +borderline of d2V +dΦ2 from negative to positive (if there is a rounded transition in a small enough +∆Φ at the bottom to connect the left and right sides of the potential, see [62]), or until the +contribution of the radiation produced in reheating process becomes non-negligible. +It is helpful to understand the behavior of oscillating inflation from the perspective of the +effective equation of state. For an oscillating scalar field Φ, its effective equation of state in +one oscillating period is defined as +⟨w⟩ ≡ ⟨p⟩ +⟨ρ⟩ = ⟨ ˙Φ2 − ρ⟩ +⟨ρ⟩ += ⟨ ˙Φ2⟩ +Vm +− 1 = ⟨Φ dV +dΦ⟩ +Vm +− 1 = 1 − 2⟨V ⟩ +Vm +, +(48) +where ⟨⟩ means the average value in one oscillation period, and Vm represents the maximal +potential of this oscillation period. +The accelerating expansion of the universe requires +⟨w⟩ < − 1 +3, which is equivalent to the following relation +U ≡ ⟨V − ΦdV +dΦ⟩ > 0. +(49) +In fact, U amounts to the intercept of the tangent to the potential at a certain Φ, shown +as the upper part of Fig. 4. As long as the intercept is positive and the contribution of +radiation is insignificant, the accelerating expansion will proceed successfully. This is the +reason why a non-convex potential can bring about oscillating inflation. +16 + +For the process with oscillating inflation, the definition of e-folding number should be +replaced to +˜N ≡ ln afHf +aiHi +≡ ln aeHe +aiHi ++ ln aoHo +aiHi +≃ N + No, +(50) +where the subscripts i and e have been defined in the last section, af and Hf represent +the cosmic scale and Hubble parameter when the full inflationary period ends, ao and Ho +represent their multiple of increase or decrease during the oscillating inflation. It indicates +that the new definition is equivalent to adding a correction No based on the e-folding number +of slow-rolling period if we take He ≈ Hi. Generally, No is related to the shape of potential +near its vacuum, reheating efficiency, and the scale of the aforementioned rounded bottom. +Given that our model does not possess an explicit rounded bottom, No depends only on +the first two aspects. +For the shape of potential, actually, our model has the following +approximate form near the center vacuum +V (Φ) ≃ ξ(Φ4 − Φ2) +2α ++ ξ2Φ4 +3α +�� +1 + +1 +ξΦ2 +�3/2 +− 1 +� +, +(51) +where ξ ≡ +α2 +3βζ. Since α determines the height of the potential, which has been fixed for +each set of ζ and β according to the observation result of ∆2 +s ∼ +V +24π2ϵ ∼ 2.1 × 10−9 [75], the +shape of the potential is essentially determined by ξ in the oscillatory region. For reheating +efficiency, we consider a constant transfer rate Γ and the transferred energy all turns to +radiation ρr +¨Φ + (3H + Γ) ˙Φ + dV +dΦ = 0, +(52) +˙ρr + 4Hρr − Γ ˙Φ2 = 0. +(53) +Then No is substantially related to the parameters ξ and Γ. +We numerically solve the above equations, and visualize in the lower part of Fig. 4. It +is transparent that if ξ ≫ 0.1, oscillating inflation will bring appreciable correction to the +e-folding number. +Because an inefficient reheating process will postpone the end of the +oscillating inflation, we can see a smaller Γ corresponds to a larger No for a certain ξ. +However, No will tend to a fixed value as Γ decreases. This property can be understood as +follows. We can prove that the potential has a quasi-linear form when Φ → 0 +V |Φ→0 ≃ +√ξ +3α |Φ|, +(54) +17 + +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +10-2 +100 +102 +104 +0 +0.5 +1 +1.5 +FIG. 4. Oscillating inflation in the center-evolving pattern of Weyl R2 + R3 model. The upper +part is a diagram for visualizing the condition of oscillating inflation, where the effective equation +of state ⟨w⟩ < − 1 +3 is equated with that the intercept U of the tangent to a certain point on the +potential corresponding to the average amplitude is positive. The lower part shows the increased +e-folding number during the oscillating inflation for various ξ and reheating efficiency Γ. +which implies that U|Φ→0 → 0 according to its definition as the intercept of the tangent to +the potential. Hence ⟨w⟩ will quickly converge to − 1 +3 as the oscillation proceeds, and No will +soon grow to a nearly constant maximum if Γ is too small to make the universe promptly +produce enough radiation to stop the oscillating inflation. This is the reason why No has an +extreme for each ξ. +Now we consider the reheating is inefficient, that is to adopt No with Γ → 0, to derive +the slow-roll e-folding number N corresponding to ˜N ∼ (50, 60), and then to calculate ns +18 + +103 +104 +105 +106 +107 +108 +10-7 +10-5 +10-3 +103 +104 +105 +106 +107 +108 +10-7 +10-5 +10-3 +<10-4 +10-3 +10-2 +FIG. 5. Possible parameter space for Weyl R2 + R3 model when Φ evolves to the center vacuum. +Here the total e-folding number ˜N ≡ N + No is considered with Γ → 0. The meaning of markers +is the same as that in Fig. 3, except for the color correspondence of r. +and r for various parameters ζ and γ. The viable parameter space is depicted in Fig. 5, +where the meaning of markers is the same as that in Fig. 3, except for the scale of color +bar. It is evident that the observation constraint on ns limits the parameters to ζ > 103 and +γ < 5 × 10−4. r has an upper limit ∼ 0.006, but no lower limit in this case. +V. +CONCLUSIONS +Cosmological observations have suggested that our universe has a nearly scaling invariant +power spectrum of the primordial density perturbation, which motivates the scaling sym- +19 + +metry as the possible feature of the underlying fundamental theories that lead to inflation. +We present the theoretical formalism of the Weyl scaling invariant gravity, ˆR2 + ˆR3. We +show this model in Eq. (1) can be rewritten equivalently to the Einstein gravity coupled +with a massive gauge boson, and a scalar field as the inflaton. We further discuss the viable +ranges of the scalar potential according to the requirement for reality and demonstrate how +the R3 term would affect the shape of potentials. Compared with the Weyl R2 inflationary +potential [41, 45] with two side minima, the R3 extension brings an additional minimum at +center. Hence, there are two viable scenarios for the inflation in this model. The first is +to roll towards the side minima, while the other is a new situation of rolling towards the +center minimum. Both scenarios allows viable parameter spaces that be probed by future +experiments on cosmic microwave background and primordial gravitational wave. +For the first scenario, we calculate the spectral index ns and tensor-to-scalar ratio r +of primordial perturbations both analytically and numerically, and contrast the parameter +spaces with the latest observational constraints. The results manifest that the level of cubic +curvature is limited to |γ| < 6×10−3, and the prediction of r in this pattern has a wide range +from O(10−4) to the upper limit of the observations, O(10−2). These results are significantly +different from the R3-extended Starobinsky model. +For the second scenario, a special process called oscillating inflation emerges after the +familiar slow-roll inflation because the potential near the center minimum is a non-convex +function that can lead to a sufficiently negative value of average equation of state. +We +calculate the correction of e-folding number in the oscillating inflation stage, and then derive +the viable parameter spaces. The results indicate that the parameters are limited to γ < +5 × 10−4 and ζ > 103. Moreover, r has an upper limit ∼ 0.006, but no lower limit. +ACKNOWLEDGMENTS +QYW and YT thank Shi Pi for helpful discussions. YT is supported by National Key Re- +search and Development Program of China (Grant No.2021YFC2201901), and Natural Sci- +ence Foundation of China (NSFC) under Grants No. 11851302. YLW is supported by the Na- +tional Key Research and Development Program of China under Grant No.2020YFC2201501, +and NSFC under Grants No. 11690022, No. 11747601, No. 12147103, and the Strategic Prior- +ity Research Program of the Chinese Academy of Sciences under Grant No. XDB23030100. +20 + +Appendix A: Analytical treatment of Starobinsky inflation +We give an analytical calculation of the tensor-to-scalar ratio r and spectral index ns in +the Starobinsky inflationary model, namely, the Einstein gravity modified by a R2 term. +The effective scalar potential can be written as +V (φ) = 1 +8α +� +1 − e−√ +2/3φ�2 +, +(A1) +where α is the coefficient of R2. The relevant two slow-roll parameters are computed as +ϵ = 4 +3 +1 +� +e +√ +2/3φ − 1 +�2, +η = −4 +3 +e +√ +2/3φ − 2 +� +e +√ +2/3φ − 1 +�2. +(A2) +Since inflation ends when ϵ ∼ 1 is reached first (η ≃ −0.15), we have +φe = +� +3 +2 ln +� +1 + 2 +√ +3 +� +≃ 0.94MP. +(A3) +Then according to Eq. (23), the e-folding number is +N = +� +3 +4 +� +e +√ +2/3φ − +� +2 +3φ +��φe +φi += 3 +4 +� +e +√ +2/3φi − e +√ +2/3φe − +� +2 +3(φi − φe) +� +. +(A4) +For N ∼ (50, 60), we find that approximately +φi ≃ +� +3 +2 ln +�4 +3(N + 4.3) +� +. +(A5) +Substituting it into Eq. (A2), we finally derive +r = 16ϵ = +12 +(N + 3.55)2, +(A6) +ns = 1 − 6ϵ + 2η = 1 − +2 +N + 3.55 − +3 +(N + 3.55)2. +(A7) +These results are shown as the red line in Fig. 2. +[1] A. H. Guth, Phys. Rev. D 23, 347-356 (1981). +[2] A. D. Linde, Phys. Lett. B 108, 389-393 (1982). +[3] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rept. 215, 203-333 (1992). +21 + +[4] Y. Akrami et al. [Planck], Astron. Astrophys. 641, A10 (2020). +[5] V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33, 532-535 (1981). +[6] S. W. Hawking, Phys. Lett. B 115, 295 (1982). +[7] A. H. Guth and S. Y. Pi, Phys. Rev. Lett. 49, 1110-1113 (1982). +[8] A. A. Starobinsky, Phys. Lett. B 117, 175-178 (1982). +[9] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D 28, 679 (1983). +[10] H. Weyl, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1918, 465 (1918). +[11] H. Weyl, Annalen Phys. 59, 101-133 (1919). +[12] L. Smolin, Nucl. Phys. B 160, 253-268 (1979). +[13] H. Cheng, Phys. Rev. Lett. 61, 2182 (1988). +[14] H. Nishino and S. Rajpoot, Phys. Rev. D 79, 125025 (2009). +[15] C. Romero, J. B. Fonseca-Neto and M. L. Pucheu, Class. Quant. Grav. 29, 155015 (2012). +[16] I. Bars, P. Steinhardt and N. Turok, Phys. Rev. D 89, no.4, 043515 (2014). +[17] I. Quiros, [arXiv:1401.2643 [gr-qc]]. +[18] E. Scholz, Gen. Rel. Grav. 47, no.2, 7 (2015). +[19] H. C. Ohanian, Gen. Rel. Grav. 48, no.3, 25 (2016). +[20] P. G. Ferreira, C. T. Hill and G. G. Ross, Phys. Rev. D 95, no.4, 043507 (2017). +[21] M. de Cesare, J. W. Moffat and M. Sakellariadou, Eur. Phys. J. C 77, no.9, 605 (2017). +[22] P. G. Ferreira, C. T. Hill and G. G. Ross, Phys. Rev. D 98, no.11, 116012 (2018). +[23] P. G. Ferreira, C. T. Hill, J. Noller and G. G. Ross, Phys. Rev. D 97, no.12, 123516 (2018). +[24] Y. Tang and Y. L. Wu, Phys. Lett. B 784, 163-168 (2018). +[25] D. M. Ghilencea and H. M. Lee, Phys. Rev. D 99, no.11, 115007 (2019) +[26] C. Wetterich, [arXiv:1901.04741 [hep-th]]. +[27] Y. Tang and Y. L. Wu, Phys. Lett. B 803, 135320 (2020). +[28] Y. Tang and Y. L. Wu, JCAP 03, 067 (2020) +[29] D. M. Ghilencea, Eur. Phys. J. C 82, no.1, 23 (2022). +[30] D. M. Ghilencea and T. Harko, [arXiv:2110.07056 [gr-qc]]. +[31] Y. L. Wu, Phys. Rev. D 93, no.2, 024012 (2016). +[32] Y. L. Wu, +Eur. Phys. J. C 78, +no.1, +28 (2018) doi:10.1140/epjc/s10052-017-5504-3 +[arXiv:1712.04537 [hep-th]]. +22 + +[33] Y. L. Wu, Int. J. Mod. Phys. A 36, no.28, 2143001 (2021) doi:10.1142/S0217751X21430016 +[arXiv:2104.05404 [physics.gen-ph]]. +[34] Y. L. Wu, Int. J. Mod. Phys. A 36, no.28, 2143002 (2021) doi:10.1142/S0217751X21430028 +[arXiv:2104.11078 [physics.gen-ph]]. +[35] D. M. Ghilencea, JHEP 03, 049 (2019). +[36] P. G. Ferreira, C. T. Hill, J. Noller and G. G. Ross, Phys. Rev. D 100, no.12, 123516 (2019). +[37] D. M. Ghilencea, JHEP 10, 209 (2019). +[38] I. Oda, [arXiv:2003.01437 [hep-th]]. +[39] D. M. Ghilencea, Eur. Phys. J. C 80, no.12, 1147. +[40] I. Oda, PoS CORFU2019, 070 (2020). +[41] Y. Tang and Y. L. Wu, Phys. Lett. B 809, 135716 (2020). +[42] I. Oda, Mod. Phys. Lett. A 35, no.37, 2050304 (2020). +[43] D. M. Ghilencea, Eur. Phys. J. C 81, no.6, 510 (2021). +[44] R. G. Cai, Y. S. Hao and S. J. Wang, Commun. Theor. Phys. 74, no.9, 095401 (2022). +[45] Q. Y. Wang, Y. Tang and Y. L. Wu, Phys. Rev. D 106, no.2, 023502 (2022). +[46] A. A. Starobinsky, Phys. Lett. B 91, 99-102 (1980). +[47] A. Vilenkin, Phys. Rev. D 32, 2511 (1985). +[48] M. B. Mijic, M. S. Morris and W. M. Suen, Phys. Rev. D 34, 2934 (1986). +[49] K. i. Maeda, Phys. Rev. D 37, 858 (1988). +[50] M. Aoki, J. Kubo and J. Yang, JCAP 01, no.01, 005 (2022). +[51] K. S. Stelle, Phys. Rev. D 16, 953-969 (1977). +[52] B. Whitt, Phys. Lett. B 145, 176-178 (1984). +[53] J. D. Barrow and S. Cotsakis, Phys. Lett. B 214, 515-518 (1988). +[54] P. A. R. Ade et al. [BICEP and Keck], Phys. Rev. Lett. 127, no.15, 151301 (2021). +[55] Q. G. Huang, JCAP 02, 035 (2014). +[56] T. Asaka, S. Iso, H. Kawai, K. Kohri, T. Noumi and T. Terada, PTEP 2016, no.12, 123E01 +(2016). +[57] D. Y. Cheong, H. M. Lee and S. C. Park, Phys. Lett. B 805, 135453 (2020). +[58] G. Rodrigues-da-Silva, J. Bezerra-Sobrinho and L. G. Medeiros, Phys. Rev. D 105, no.6, +063504 (2022). +[59] V. R. Ivanov, S. V. Ketov, E. O. Pozdeeva and S. Y. Vernov, JCAP 03, no.03, 058 (2022). +23 + +[60] Y. Shtanov, V. Sahni and S. S. Mishra, [arXiv:2210.01828 [gr-qc]]. +[61] T. Modak, L. R¨over, B. M. Sch¨afer, B. Schosser and T. Plehn, [arXiv:2210.05698 [astro- +ph.CO]]. +[62] T. Damour and V. F. Mukhanov, Phys. Rev. Lett. 80, 3440-3443 (1998). +[63] A. R. Liddle and A. Mazumdar, Phys. Rev. D 58, 083508 (1998). +[64] A. Taruya, Phys. Rev. D 59, 103505 (1999). +[65] V. H. Cardenas and G. Palma, Phys. Rev. D 61, 027302 (2000). +[66] J. w. Lee, S. Koh, C. Park, S. J. Sin and C. H. Lee, Phys. Rev. D 61, 027301 (2000). +[67] V. Sahni and L. M. Wang, Phys. Rev. D 62, 103517 (2000). +[68] S. Tsujikawa, Phys. Rev. D 61, 083516 (2000). +[69] M. Sami, Grav. Cosmol. 8, 309-312 (2003). +[70] S. Dutta and R. J. Scherrer, Phys. Rev. D 78, 083512 (2008). +[71] M. C. Johnson and M. Kamionkowski, Phys. Rev. D 78, 063010 (2008). +[72] H. Mohseni Sadjadi and P. Goodarzi, Phys. Lett. B 732, 278-284 (2014). +[73] J. A. R. Cembranos, A. L. Maroto and S. J. N´u˜nez Jare˜no, JHEP 03, 013 (2016). +[74] P. Goodarzi and H. Mohseni Sadjadi, Eur. Phys. J. C 77, no.7, 463 (2017). +[75] N. Aghanim et al. [Planck], Astron. Astrophys. 641, A6 (2020). +24 + diff --git a/39E2T4oBgHgl3EQfOAbJ/content/tmp_files/load_file.txt b/39E2T4oBgHgl3EQfOAbJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c9bdaf63d21a12cf843173dcd96ad28ed59f4f2 --- /dev/null +++ b/39E2T4oBgHgl3EQfOAbJ/content/tmp_files/load_file.txt @@ -0,0 +1,926 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf,len=925 +page_content='Inflation in Weyl Scaling Invariant Gravity with R3 Extensions Qing-Yang Wanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Yong Tanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='c,' metadata={'source': 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+page_content=' China bSchool of Fundamental Physics and Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hangzhou Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' UCAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hangzhou 310024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' China cInternational Center for Theoretical Physics Asia-Pacific,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Beijing/Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' China dNational Astronomical Observatories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Beijing 100101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' China eInstitute of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' China (Dated: January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2023) Abstract The cosmological observations of cosmic microwave background and large-scale structure indicate that our universe has a nearly scaling invariant power spectrum of the primordial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, the exact origin for this primordial spectrum is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Here, we propose the Weyl scaling invariant R2 + R3 gravity that gives rise to inflation that is responsible for the primordial perturbation in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We develop both analytic and numerical treatments on inflationary observables, and find this model gives a distinctive scalar potential that can support two different patterns of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The first one is similar to that occurs in the pure R2 model, but with a wide range of tensor-to-scalar ratio r from O(10−4) to O(10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The other one is a new situation with not only slow-roll inflation but also a short stage of oscillation-induced accelerating expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Both patterns of inflation have viable parameter spaces that can be probed by future experiments on cosmic microwave background and primordial gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='03744v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='CO] 10 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' INTRODUCTION Inflation is a hypothetical epoch of exponential expansion introduced in the very early universe to solve the cosmological horizon and flatness problems [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is also a reasonable scheme to explain the origin of primordial density perturbations, which plays the role of the seeds that formed the structure of current universe [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In recent years, the precise measurement of cosmic microwave background (CMB) presents us with an almost scale invariant spectrum of primordial perturbations [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This result is usually explained by an approximate de Sitter spacetime of the very early universe [5–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Moreover, it is theoretically explored that there is a more profound and basic principle behind the phenomenon, namely, local Weyl scaling invariance of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This symmetry is first proposed by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Weyl in the attempt of understanding gravity and electromagnetism in a unified framework [10, 11], and after a century of development, it has been applied extensively to particle physics, cosmology [12–30] and gauge theory of gravity [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lately, inflation in the Weyl scaling invariant theory of gravity, especially induced by a quadratic curvature term R2, has been of many concern [35–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Comparing with the conventional R2 model, which is also called Starobinsky model [46–49], the scaling invariant version not only allows a viable inflation scenario with good observational agreement, but also provides a framework to comprehend another fundamental puzzles, such as hierarchy problem [37, 40, 50] and dark matter candidates [41, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, inflation with only quadratic scalar curvature might be just a simplistic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' From the viewpoint of effective field theory, any higher-order curvature effects may exist and play a role in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hence it is reasonable to evaluate their impacts on inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Generally, the extensions with high-order tensors, like RµνRµν or RµνρσRµνρσ, can result in unacceptable ghost degrees of freedom [51], while the terms of arbitrary functions of the Ricci scalar are known to be safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore, in this paper, we consider a minimal extension of Ricci scalar beyond the R2 model with Weyl scaling invariance, namely a cubic term coupled with an extra scalar field as denominator R3/ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We will show that even if this term is extremely small, it will have an essential impact on inflation, which even open up a completely different inflationary scenario from Weyl R2 and conventional R2 + R3 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' II, we develop the analytic formalism of Weyl R2 + R3 model and derive the effective scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We show that in some cases, the 2 potential has two different kinds of global minima, leading to two distinctive inflationary pat- terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' III, we investigate the inflation in the pattern of evolving to the side minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We calculate the spectral index ns and tensor-to-scalar ratio r of the inflationary perturba- tions, and give the preferred parameter space allowed by the latest observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Analytical treatments are developed for more transparent, physical understanding of the asymptotic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Then in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' IV, we investigate the pattern of evolving to the center minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A special process called “oscillating inflation” is considered in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Finally, conclusions are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We adopt the following conventions: metric ηµν = (−1, +1, +1, +1), natural unit ℏ = c = 1 and MP ≡ 1/ √ 8πG = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='435 × 1018 GeV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' WEYL SCALING INVARIANT R2 + R3 MODEL We start with the following Lagrangian for metric field gµν, scalar field ϕ, and Weyl gauge field Wµ ≡ gWwµ with local scaling symmetry L √−g = 1 2 � ϕ2 ˆR + α ˆR2 + β ϕ2 ˆR3 � − ζ 2DµϕDµϕ − 1 4g2 W FµνF µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (1) Here g is the determinant of metric, α, β and ζ are constant parameters, Dµ = ∂µ − Wµ is the covariant derivative associated with scaling symmetry, gW is the coupling constant, Fµν ≡ ∂µWν − ∂νWµ defines the invariant field strength of Wµ, and ˆR is the Ricci scalar defined by the local scaling invariant connection ˆΓρ µν = 1 2gρσ [(∂µ + 2Wµ)gσν + (∂ν + 2Wν)gµσ − (∂σ + 2Wσ)gµν] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (2) Explicit calculation shows the relation between ˆR and usual R defined by metric field gµν, ˆR = R − 6WµW µ − 6 √−g∂µ(√−gW µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (3) It is straightforward to verify the invariance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (1) under the following Weyl scaling transformation metric : gµν → g′ µν = f 2(x)gµν, scalar : φ → φ′ = f −1(x)φ, Ricci scalar : ˆR → ˆR′ = f −2(x) ˆR, Weyl vector : Wµ → W ′ µ = Wµ − ∂µ ln f(x), (4) where f(x) is an arbitrary positive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 3 The purpose to explore the Lagrangian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (1) is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Theoretically, such a ˆR3 term constitutes as a simple extension of the ˆR2 theory, motivated from perspective of effective field theories and also quantum loop corrections in more fundamental theories [31– 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phenomenologically, it is worthwhile to explore how such a term would modify the cosmological observations related to inflation, and evaluate the likelihood and robustness of the predictions in the lowest-order theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Formalism in Einstein frame General f(R) gravity is equivalent to the Einstein gravity with a scalar field [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [41], we have extended the proof in general scaling invariant F( ˆR, ϕ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can explicitly show that by introducing an auxiliary scalar field χ and rewrite the high-order curvature terms as F( ˆR, ϕ) ≡ ϕ2 ˆR + α ˆR2 + β ϕ2 ˆR3 = F ˆR( ˆR → χ2, ϕ)( ˆR − χ2) + F( ˆR → χ2, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (5) Here F ˆR denotes the derivative over ˆR, F ˆR = ∂F( ˆR, ϕ)/∂ ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can verify that the equiv- alence relation χ2 = ˆR can be obtained from the Euler-Lagrange equation, δL δχ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Substi- tuting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (5) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (1), we find L √−g = 1 2 � ϕ2 + 2αχ2 + 3β ϕ2 χ4 � ˆR − 1 2 � αχ4 + 2β ϕ2 χ6 � − ζ 2DµϕDµϕ − 1 4g2 W FµνF µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (6) Now we have demonstrated that linearization of ˆR has led to the non-minimal coupling of the scalar field, χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can transform the above Lagrangian into the Einstein frame by making a Weyl or conformal transformation of the metric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, we note that scaling invariance is still preserved in our model with χ → χ′ = f −1(x)χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore, we can directly normalize the coefficient before the Ricci scalar as ϕ2 + 2αχ2 + 3βχ4/ϕ2 = 1, (7) due to the scaling invariance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This is equivalent to making a Weyl transformation with f(x) = � ϕ2 + 2αχ2 + 3βχ4/ϕ2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Further dropping the total derivative term 4 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (3) due to its null surface integral, we can write the Lagrangian as L √−g =1 2R − ζ 2DµϕDµϕ − V (ϕ) − 1 4g2 W FµνF µν − 3W µWµ =R 2 − ∂µϕ∂µϕ 2/ζ + ϕ2/3 − V (ϕ) − 1 4g2 W FµνF µν − 6 + ζϕ2 2 � Wµ − ∂µ ln |6 + ζϕ2| 2 �2 , (8) with the scalar potential V (ϕ) = α 2 χ4 + β ϕ2χ6 = α 6β � ϕ4 − ϕ2� + α3ϕ4 27β2 �� 1 − 3β α2 � 1 − ϕ−2��3/2 − 1 � , (9) where we have solved χ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (7) χ2 = αϕ2 3β �� 1 − 3β α2 (1 − ϕ−2) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (10) It is now clear that we have a minimally-coupled scalar ϕ with a non-canonical kinetic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' To further simplifying the theoretical formalism, we introduce the following redefini- tions for the scalar and the Weyl gauge field ϕ2 ≡ � � � � � 6 |ζ| sinh2 � ±Φ √ 6 � for ζ > 0, 6 |ζ| cosh2 � ±Φ √ 6 � for ζ < 0, (11) ˜Wµ ≡ Wµ − 1 2∂µ ln |6 + ζϕ2| ≡ gW ˜wµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (12) Then the final Lagrangian turns into a more compact form L √−g = 1 2R − 1 2∂µΦ∂µΦ − V (Φ) − 1 4g2 W ˜Fµν ˜F µν − 1 2m2(Φ) ˜W µ ˜Wµ, (13) with the mass term of Weyl gauge field m2(Φ) = � � � � � +6 cosh2 � Φ √ 6 � for ζ > 0, −6 sinh2 � Φ √ 6 � for ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (14) We should note that m2(Φ) is negative when ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore, to avoid Weyl gauge boson becoming tachyonic in this case, it requires some other mechanisms to obtain a real mass, for example, introducing other scalar field, which we do not explore in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For viable inflation, both positive and negative are possible, as we shall show later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In the above discussion, we have demonstrated that Weyl scaling invariant ˆR2+ ˆR3 model can be written equivalently as the Einstein gravity coupled with a self-interacting scalar Φ 5 and a massive vector ˜Wµ with a field-dependent mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This conclusion is also true for any Weyl scaling invariant model of gravity with high-order curvature ˆRn as the above formalism applies straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is also worth to point out that Weyl vector boson can serve as a dark matter candidate [27, 28, 41], with details of the relic abundance being discussed in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In this paper, we shall concentrate on the scalar potential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (9) and discuss the viable inflation scenarios with the presence of ˆR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Effective scalar potentials There are two necessary requirements for the potential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The first one is ϕ2 > 0 since ϕ is a real scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The other is 1 − 3β α2 � 1 − 1 ϕ2 � ≥ 0, otherwise an imaginary potential will emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Consequently, there are some constraints on the parameters and the viable value of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can rewrite the second requirement as sinh �±Φ √ 6 � ≥ or ≤ � |ζ| 6 − 2α2/β , for ζ > 0, cosh �±Φ √ 6 � ≥ or ≤ � |ζ| 6 − 2α2/β , for ζ < 0, (15) where “ ≥ ” for β < α2 3 and “ ≤ ” for β ≥ α2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For convenience, we define λ ≡ � |ζ| 6−2α2/β and γ ≡ 3β α2, then discuss the possible ranges of the potential corresponding to different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The results are listed in the Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' To ensure the theoretical stability, we require that Φ can only evolve within these ranges where the potential is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 1 shows some instances of the scalar potential for several values of ζ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We first discuss the case of positive ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' When γ = 0, it is a hill-top-like potential with two minima at Φ = ± √ 6 sinh−1 � ζ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, as long as there is a tiny cubic curvature, whether positive or negative, the shape of potential will be affected significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' When γ > 0, the potential turns to decrease near Φ = 0, and a third vacuum can form there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This behavior is transparent, because when ζ > 0, Φ = 0 corresponds to ϕ2 = 0 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (11), then substituting it in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (9) will obtain V |Φ=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' When γ < 0, the potential turns to rise near Φ = 0 and become imaginary and unphysical in − √ 6 sinh−1 λ < Φ < √ 6 sinh−1 λ, which has been listed in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Next, we switch to the case of negative ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 1 that when ζ < 0 and |ζ| or |γ| is relatively small, the modification of ˆR3 term on the Weyl R2 potential is moderate, 6 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Effective potential range of the Weyl R2 + R3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' ζ γ or β real V (ϕ) ζ > 0 γ ≥ 1 |Φ| ≤ √ 6 sinh−1 λ 0 ≤ γ < 1 fully real γ < 0 |Φ| ≥ √ 6 sinh−1 λ −6 < ζ < 0 γ > 1 1+ζ/6 fully imaginary 1 < γ ≤ 1 1+ζ/6 |Φ| ≤ √ 6| cosh−1 λ| γ ≤ 1 fully real ζ ≤ −6 γ ≥ 1 |Φ| ≤ √ 6| cosh−1 λ| 1 1+ζ/6 < γ < 1 fully real γ ≤ 1 1+ζ/6 |Φ| ≥ √ 6| cosh−1 λ| unlike the dramatic change near Φ = 0 in the case of positive ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This is because the mapping of Φ ⇒ ϕ2 does not cover the interval of ϕ2 < 1 for ζ < 0 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In other words, for negative ζ with modest |γ|, Φ → 0 does not lead to ϕ2 → 0, which brings the violent behavior of the potential around here in the case of ζ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, when ζ is excessively negative or |γ| is large enough, the violent variation will reappear to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For γ > 0, the potential will return to a downward trend near Φ = 0, albeit there is no true vacuum formed (but a false vacuum is formed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' And for excessively negative γ, the imaginary potential will reappear in the range of − √ 6| cosh−1 λ| < Φ < √ 6| cosh−1 λ|, which we have listed this situation in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' I (see ζ ≤ −6 with γ ≤ 1 1+ζ/6 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Generally, inflation takes place when the potential is flat and Φ evolves to the vacuum (Φ|V =0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The cosmological observations would restrict the potential and the initial value Φi when inflation starts, here the Φi is defined as the value when the comoving horizon of the inflationary universe shrinks to the same size as today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For ζ > 0 and γ > 0, the scalar potential contains three separate vacua, one lying at the center and the other two at both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore, there are two different viable inflationary patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' One pattern refers to the evolution into the central minimum, and the other into the side minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can calculate the value of Φ which corresponds to the hill-top of the 7 10 5 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 2 10-10 10 5 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 2 10-10 10 5 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 2 10-10 10 5 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 2 10-10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Effective potentials of Weyl R2 + R3 model with α = 109 and various γ and ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Here we only depict the real ranges of potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' potential in this case Φh = ± √ 6 sinh−1 � ζ 12 √3γ − 2γ 3 − 4γ , (16) which is the critical point of two inflationary patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Neglecting the velocity, if the initial value of inflation field satisfies |Φi| > |Φh|, it will evolve towards the side vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' If |Φi| < |Φh| at the beginning, the inflation field will evolve towards the central vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For other cases of ζ and γ, there are only the global side minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hence the only feasible inflationary pattern is that Φ evolves to either one of the side minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The initial value Φi has to correspond to a real potential, and when there is a false vacuum in ζ < 0 case, it requires a large enough |Φi| outside two local maxima of the potential to ensure the gradient of V (Φi) towards the true vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Next, we are going to discuss the inflation in these two patterns respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 8 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' INFLATION TO THE SIDE In this inflation pattern, ϕ2 (defined as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (11)) is usually not very close to 0, and as we shall show later, observations generally would require an extremely small cubic curva- ture, namely |γ| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore in many cases, |γ(1 − ϕ−2)| ≪ 1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Under this condition, we are able to have analytical treatment and expand the potential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (9) as V (ϕ) =ϕ4 − ϕ2 2αγ + ϕ4 3αγ2 � −3γ 2 � 1 − 1 ϕ2 � + 3γ2 8 � 1 − 1 ϕ2 �2 + γ3 16 � 1 − 1 ϕ2 �3 + O �γ4 ϕ8 �� = 1 8α � 1 − ϕ2�2 � 1 + γ 6 � 1 − 1 ϕ2 � + O �γ2 ϕ4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (17) Then with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (11), we derive V (Φ) = � � � � � 1 8α � 1 − 6 |ζ| sinh2 � Φ √ 6 ��2 � 1 + γ 6 � 1 − |ζ| 6 csch2 � Φ √ 6 �� + O(γ2) � for ζ > 0, 1 8α � 1 − 6 |ζ| cosh2 � Φ √ 6 ��2 � 1 + γ 6 � 1 − |ζ| 6 sech2 � Φ √ 6 �� + O(γ2) � for ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (18) The first term is exactly the effective potential of Weyl ˆR2, which has been shown in [41, 45], and the rest originates from the cubic curvature term ˆR3, to the leading order of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Next we shall calculate the inflationary physical quantities, the spectral index ns and tensor-to- scalar ratio r, and contrast them with the latest observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We first give an analytical calculation for two limiting cases, then show the full numerical results for general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Analytical approach of γ → 0 case We first discuss the γ → 0 case and show how ζ affects ns and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The slow-roll parameters in this case can be derived as ϵ ≡ 1 2 �V ′(Φ) V �2 = 12 sinh2 � 2Φ √ 6 � � |ζ + 3| − 3 − 6 sinh2 � Φ √ 6 ��2, (19) η ≡ V ′′(Φ) V = 12 cosh � 4Φ √ 6 � − 4|ζ + 3| cosh � 2Φ √ 6 � � |ζ + 3| − 3 − 6 sinh2 � Φ √ 6 ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (20) Generally, the slow-roll inflation occurs when ϵ and |η| is small enough, and it will end when any of them evolves to ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For the situation we are concerned with, ϵ breaks the slow-roll 9 limit before the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Thus we derive the value of Φ when inflation ends according to ϵ = 1 Φe = � 3 2 ln � 2 � |ζ + 3|2 + 3 √ 3 − |ζ + 3| + � 7 3|ζ + 3|2 − 4|ζ + 3| √ 3 � |ζ + 3|2 + 3 + 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (21) When |ζ| > O(102), which is a preferred range by the observational constraints as we will show shortly, the above equation can be approximated as Φe ≃ � 3 2 ln � 1 √ 3 � 2 + � 7 − 4 √ 3 − √ 3 � |ζ + 3| � ≃ � 3 2 ln (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3094|ζ + 3|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (22) It is now clear that when |ζ| is large enough, Φe will be almost independent of the sign of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Next, we calculate initial value Φi, which is defined when the size of comoving horizon during inflation shrinks to the present size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We first focus on the e-folding number of the slow-roll inflation N ≡ ln ae ai ≃ � Φe Φi dΦ √ 2ϵ, (23) where ai/e ≡ a(Φi/e) is the cosmic scale factor when inflation starts/ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (19) into it, we find N = (|ζ + 3| − 3) ln � tanh � Φ √ 6 �� − 6 ln � cosh � Φ √ 6 �� 4 ����� Φe Φi = |ζ + 3| − 3 4 ln � �tanh � 1 2 ln(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3094|ζ + 3|) � tanh � Φi √ 6 � � � − 3 2 ln � �cosh � 1 2 ln(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3094|ζ + 3|) � cosh � Φi √ 6 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (24) For the circumstances we are concerned with, namely N ∼ (50, 60) and |ζ| > O(102), the second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (24) is much smaller than the first term, and it can be estimated as ∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Thus we derive Φi ≃ √ 6 tanh−1 �� 1 − 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3094|ζ + 3| + 1 � e −4(N+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3) |ζ+3|−3 � ≡ √ 6 tanh−1 Ω(ζ, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (25) Here we have defined Ω(ζ, N) for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' When |ζ| ≫ 4N, it can be further approximated as Φi ≃ � 3 2 ln |ζ| 2N+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (25) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (19) and (20), we find ϵi = 48Ω2 [(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2, (26) ηi =4 [(Ω4 − 1)|ζ + 3| + 3(Ω4 + 6Ω2 + 1)] [(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (27) 10 As a result, the tensor-to-scalar ratio r and spectral index ns of inflationary perturbations in the γ → 0 limit are finally calculated as r = 16ϵi = 768Ω2 [(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2, (28) ns = 1 − 6ϵi + 2ηi = 1 + 8(Ω4 − 1)|ζ + 3| + 24(Ω4 − 6Ω2 + 1) [(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (29) For N ∼ (50, 60) and |ζ| > O(102), We can approximate the expressions as r ≃ r∗ − 54 ζ2 , (30) ns ≃ n∗ s − 11N ζ2 , (31) where r∗ ≃ 12 (N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55)2, n∗ s ≃ 1 − 2 N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55 − 3 (N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55)2 (32) are the predictions of Starobinsky model (see Appendix A for an analytical derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Thus it is evident that the predictions of inflationary perturbations in our model will converge to that of Starobinsky model when γ → 0 and ζ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' As |ζ| decreases, the value of r and ns will also decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We show this trend as the pink area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' According to the latest observation [54], the lower limit of ns has been constrained to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='959, hence it requires |ζ| > 270 in this γ → 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Analytical approach of ζ → ∞ case Now we discuss the ζ → ∞ case and show how γ affects r and ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' When ζ is large enough, the potential is greatly widened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The side vacua are far away from 0 and so are Φi and Φe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=', Φi ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4MP, Φe ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='8MP for ζ = 104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (11) can be approximated as ϕ2 = 6 |ζ| � eΦ/ √ 6 ± e−Φ/ √ 6 2 �2 ≃ e √ 2/3 � Φ−√ 3/2 ln(2|ζ|/3) � ≡ e √ 2/3(Φ−Φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (33) Here and after, without losing generality, we may choose to evolve in the positive Φ region, and denote Φ0 as the minimum in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (17), we have the scalar potential for Φ ≫ 0 V (Φ) = 1 8α � 1 − e √ 2/3(Φ−Φ0)�2 � 1 + γ 6 � 1 − e−√ 2/3(Φ−Φ0)� + O(γ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (34) 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The predictions of spectral index ns combined with tensor-to-scalar ratio r in the Weyl R2 + R3 model with e-folding number N ∼ (50, 60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The pink area shows the results in the γ → 0 case with various ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The yellow and green areas respectively show the ζ → ∞ and ζ = −650 cases with various γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The red line is the result with both γ → 0 and ζ → ∞, which is equivalent to the Starobinsky model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The blue area is the latest observation constraint given by the BICEP/Keck collaboration [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ignoring the O(γ2) terms, we give an approximate expression for the slow-roll parameters ϵ ≡ 1 2 �V ′(Φ) V �2 ≃ � γe √ 2/3(Φ−Φ0) − 2(γ + 6)e √ 8/3(Φ−Φ0) + γ �2 3 � e √ 2/3(Φ−Φ0) − 1 �2 � γ − (γ + 6)e √ 2/3(Φ−Φ0)�2, (35) η ≡ V ′′(Φ) V ≃ 6(γ + 4)e √ 8/3(Φ−Φ0) − 8(γ + 6)e √ 6(Φ−Φ0) + 2γ 3 � e √ 2/3(Φ−Φ0) − 1 �2 � γ − (γ + 6)e √ 2/3(Φ−Φ0)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (36) In this case, the slow-roll inflation also ends at ϵ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' To find the expression of Φe, we further approximate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (35) as ϵ ≃ e−√ 8/3(Φ−Φ0) � γ − 12e √ 8/3(Φ−Φ0)�2 108 � e √ 2/3(Φ−Φ0) − 1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (37) 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' → 60= -650 95% CL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='03 68% CL 5×1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='01 500 X = 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='003 3x 10 3 × 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='975 ns0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1 0← N = 50 8个VThen Φe can be derived as Φe = Φ0 − � 3 2 ln �√ 3 γ �� 2(2 + √ 3)γ + 9 − 3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (38) If γ is extremely small, we will find Φe ≃ Φ0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='94MP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Next, we derive the analytic formula for Φi in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The e-folding number of the slow-roll inflation can be calculated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (37) as N = − �27 4γ tanh−1 �� γ 12e−√ 2 3 (Φ−Φ0) � − 3 8 ln � 12 − γe−√ 8 3 (Φ−Φ0)� − √ 6 4 (Φ − Φ0) ���� Φe Φi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (39) Considering N ∼ (50, 60) and γ < O(10−3), the first term of the integral is dominant, while the rest are the marginal terms which can be approximately treated as a constant, ∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hence we have N ≃ �27 4γ � tanh−1 �� γ 12e−√ 2/3(Φi−Φ0) � − tanh−1 �� γ 12e−√ 2/3(Φe−Φ0) �� − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='7, (40) and derive Φi = Φ0 − � 3 2 ln ����� �12 γ tanh � tanh−1 �� γ 12e−√ 2/3(Φe−Φ0) � + � 4γ 27(N + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='7) ������ ≃ Φ0 − � 3 2 ln ����� �12 γ tanh � tanh−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='622√γ) + � 4γ 27(N + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='7) ������ ≡ Φ0 − � 3 2 ln Θ(γ, N), (41) where we have defined Θ(γ, N) for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Then substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (35) and (36), we find ϵi = [γΘ(1 + Θ) − 2(γ + 6)]2 3 [1 − Θ]2 [γΘ − (γ + 6)]2, (42) ηi = 2γΘ3 + 6(γ + 4)Θ − 8(γ + 6) 3 [1 − Θ]2 [γΘ − (γ + 6)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (43) Finally, we derive r and ns of the inflationary perturbations in the ζ → ∞ limit r = 16ϵi = 16 [γΘ(1 + Θ) − 2(γ + 6)]2 3 [1 − Θ]2 [γΘ − (γ + 6)]2 , (44) ns = 1 − 6ϵi + 2ηi = 1 − 2 [γΘ(1 + Θ) − 2(γ + 6)]2 [1 − Θ]2 [γΘ − (γ + 6)]2 + 4γΘ3 + 3(γ + 4)Θ − 4(γ + 6) 3 [1 − Θ]2 [γΘ − (γ + 6)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (45) 13 If γ is extremely small, smaller than O(10−4), the above expressions can be linearly approx- imated as r ≃ r∗ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4γ, (46) ns ≃ n∗ s − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='42γN, (47) where r∗ and n∗ s have been defined in the last paragraph of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can see that compared with the predictions of Starobinsky model, a positive γ will reduce both r and ns, while a negative γ will increase them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We show this trend as the yellow area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is manifest that the observations have constrained |γ| ≲ 5 × 10−4 in this ζ → ∞ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Actually, this result agrees with other numerical investigations of the R3-extended Starobin- sky model [55–61], since the potential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (34) is the same as the R3-extended Starobinsky model with a vacuum shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Moreover, compared with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (30) and (31), we note that the predictions of r and ns in the γ → 0 case is similar to that of the ζ → ∞ and γ > 0 case with a simple replacement of γ ↔ 24 ζ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This can be seen more clearly from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2, where the pink area overlaps with the yellow area with γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' General cases Now we discuss the general cases with various ζ and γ by numerical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Here the parameter ranges satisfying observational constraints (see blue area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2) are marked with colored areas, where the color gradient from blue to red corresponds to ascending value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The gray areas represent that the potential defined by these parameters cannot support an adequate inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In other words, their maximal e-folding number is unable to reach N = 50 or 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The white areas are the parameter ranges that can give rise to ample inflation, but their prediction of ns or r has been excluded by the observation constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Here we mark two dotted lines to distinguish the boundaries of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Beyond the pink one indicates a large ns that exceeds the observational upper limit, while beyond the green one signifies a too small prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Let us focus on the colored parameter ranges that are allowed by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In the |ζ| ≫ 1000 case, the result is roughly equivalent to the analytical calculation shown in the last subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The prediction of r is limited to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='002 < r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, distinctive situations appear when |ζ| is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' First, when −1000 < ζ < −200, the restrictions on γ is relaxed, which can stand |γ| ∼ 6 × 10−3 at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Besides, the upper limit of r is 14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Possible parameter space for Weyl R2 + R3 model when Φ evolves to the side vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The colored areas are the parameter ranges allowed by the latest observations of BICEP/Keck collaboration [54], where the color gradient from blue to red corresponds to r increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='001 to the observational upper limit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The dotted lines are the boundaries that ns exceeds the observational upper (pink line) or lower (green line) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The gray areas represent the parameter ranges with inadequate inflation, namely, the maximal e-folding number of inflation cannot reach N = 50 or 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' greatly expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' There is even a small parameter range that gives r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We show an example as the green area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It clearly shows a distinguishable feature from the Weyl R2 model and the R3-extended Starobinsky model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' If the next generation experiment of CMB B-mode polarization detects the primordial gravitational waves with r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='01, it may support Weyl R2 + R3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Another notable feature emerges at 0 < ζ < 200, where the 15 r0 2 ns > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='974 4 inadequate e-folds 6 3000 2000 1000 0 1000 2000 × 10-3 4 N = 60 2 ns < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='959 0 2 ns > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='974 4 inadequate e-folds 9- 3000 2000 1000 0 1000 20000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='015 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='001 3000×10-3 4 N = 50 2 ns < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='959negative γ, even if very small, can greatly affect the predictions of primordial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Actually, there are some cases with small positive ζ and small negative γ can give proper r and ns that match the observation constraints, and generally, r is extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For instance, when ζ = 80, γ = −4 × 10−8, and N = 60, we have ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='963 and r = 3 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' INFLATION TO THE CENTER As we mentioned earlier, the third vacuum appears at Φ = 0 in the case of ζ > 0 and γ > 0, and if the initial value satisfies |Φi| < |Φh| (Φh is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (16)), inflation can happen in the evolution of Φ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Actually, the situation is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A process called “oscillating inflation” [62–74] will continue immediately after the end of slow-roll inflation because the scalar potential in this case is a non-convex function in the region close to the vacuum, which means there is d2V dΦ2 < 0 when Φ nears 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' for such a non-convex potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' despite the slow-roll conditions (ϵ ≪ 1 and |η| ≪ 1) has been violated during the bottom oscillation of the inflaton potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' the universe can keep accelerating expansion until the average amplitude of the inflaton’s oscillation becomes lower than the borderline of d2V dΦ2 from negative to positive (if there is a rounded transition in a small enough ∆Φ at the bottom to connect the left and right sides of the potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' see [62]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' or until the contribution of the radiation produced in reheating process becomes non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is helpful to understand the behavior of oscillating inflation from the perspective of the effective equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For an oscillating scalar field Φ, its effective equation of state in one oscillating period is defined as ⟨w⟩ ≡ ⟨p⟩ ⟨ρ⟩ = ⟨ ˙Φ2 − ρ⟩ ⟨ρ⟩ = ⟨ ˙Φ2⟩ Vm − 1 = ⟨Φ dV dΦ⟩ Vm − 1 = 1 − 2⟨V ⟩ Vm , (48) where ⟨⟩ means the average value in one oscillation period, and Vm represents the maximal potential of this oscillation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The accelerating expansion of the universe requires ⟨w⟩ < − 1 3, which is equivalent to the following relation U ≡ ⟨V − ΦdV dΦ⟩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (49) In fact, U amounts to the intercept of the tangent to the potential at a certain Φ, shown as the upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' As long as the intercept is positive and the contribution of radiation is insignificant, the accelerating expansion will proceed successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This is the reason why a non-convex potential can bring about oscillating inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 16 For the process with oscillating inflation, the definition of e-folding number should be replaced to ˜N ≡ ln afHf aiHi ≡ ln aeHe aiHi + ln aoHo aiHi ≃ N + No, (50) where the subscripts i and e have been defined in the last section, af and Hf represent the cosmic scale and Hubble parameter when the full inflationary period ends, ao and Ho represent their multiple of increase or decrease during the oscillating inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It indicates that the new definition is equivalent to adding a correction No based on the e-folding number of slow-rolling period if we take He ≈ Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Generally, No is related to the shape of potential near its vacuum, reheating efficiency, and the scale of the aforementioned rounded bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Given that our model does not possess an explicit rounded bottom, No depends only on the first two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For the shape of potential, actually, our model has the following approximate form near the center vacuum V (Φ) ≃ ξ(Φ4 − Φ2) 2α + ξ2Φ4 3α �� 1 + 1 ξΦ2 �3/2 − 1 � , (51) where ξ ≡ α2 3βζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Since α determines the height of the potential, which has been fixed for each set of ζ and β according to the observation result of ∆2 s ∼ V 24π2ϵ ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1 × 10−9 [75], the shape of the potential is essentially determined by ξ in the oscillatory region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For reheating efficiency, we consider a constant transfer rate Γ and the transferred energy all turns to radiation ρr ¨Φ + (3H + Γ) ˙Φ + dV dΦ = 0, (52) ˙ρr + 4Hρr − Γ ˙Φ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (53) Then No is substantially related to the parameters ξ and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We numerically solve the above equations, and visualize in the lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is transparent that if ξ ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1, oscillating inflation will bring appreciable correction to the e-folding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Because an inefficient reheating process will postpone the end of the oscillating inflation, we can see a smaller Γ corresponds to a larger No for a certain ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, No will tend to a fixed value as Γ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This property can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can prove that the potential has a quasi-linear form when Φ → 0 V |Φ→0 ≃ √ξ 3α |Φ|, (54) 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4 10-2 100 102 104 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Oscillating inflation in the center-evolving pattern of Weyl R2 + R3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The upper part is a diagram for visualizing the condition of oscillating inflation, where the effective equation of state ⟨w⟩ < − 1 3 is equated with that the intercept U of the tangent to a certain point on the potential corresponding to the average amplitude is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The lower part shows the increased e-folding number during the oscillating inflation for various ξ and reheating efficiency Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' which implies that U|Φ→0 → 0 according to its definition as the intercept of the tangent to the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hence ⟨w⟩ will quickly converge to − 1 3 as the oscillation proceeds, and No will soon grow to a nearly constant maximum if Γ is too small to make the universe promptly produce enough radiation to stop the oscillating inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This is the reason why No has an extreme for each ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Now we consider the reheating is inefficient, that is to adopt No with Γ → 0, to derive the slow-roll e-folding number N corresponding to ˜N ∼ (50, 60), and then to calculate ns 18 103 104 105 106 107 108 10-7 10-5 10-3 103 104 105 106 107 108 10-7 10-5 10-3 <10-4 10-3 10-2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Possible parameter space for Weyl R2 + R3 model when Φ evolves to the center vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Here the total e-folding number ˜N ≡ N + No is considered with Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The meaning of markers is the same as that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 3, except for the color correspondence of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' and r for various parameters ζ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The viable parameter space is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 5, where the meaning of markers is the same as that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 3, except for the scale of color bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is evident that the observation constraint on ns limits the parameters to ζ > 103 and γ < 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' r has an upper limit ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='006, but no lower limit in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' CONCLUSIONS Cosmological observations have suggested that our universe has a nearly scaling invariant power spectrum of the primordial density perturbation, which motivates the scaling sym- 19 metry as the possible feature of the underlying fundamental theories that lead to inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We present the theoretical formalism of the Weyl scaling invariant gravity, ˆR2 + ˆR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We show this model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (1) can be rewritten equivalently to the Einstein gravity coupled with a massive gauge boson, and a scalar field as the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We further discuss the viable ranges of the scalar potential according to the requirement for reality and demonstrate how the R3 term would affect the shape of potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Compared with the Weyl R2 inflationary potential [41, 45] with two side minima, the R3 extension brings an additional minimum at center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hence, there are two viable scenarios for the inflation in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The first is to roll towards the side minima, while the other is a new situation of rolling towards the center minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Both scenarios allows viable parameter spaces that be probed by future experiments on cosmic microwave background and primordial gravitational wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For the first scenario, we calculate the spectral index ns and tensor-to-scalar ratio r of primordial perturbations both analytically and numerically, and contrast the parameter spaces with the latest observational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The results manifest that the level of cubic curvature is limited to |γ| < 6×10−3, and the prediction of r in this pattern has a wide range from O(10−4) to the upper limit of the observations, O(10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' These results are significantly different from the R3-extended Starobinsky model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For the second scenario, a special process called oscillating inflation emerges after the familiar slow-roll inflation because the potential near the center minimum is a non-convex function that can lead to a sufficiently negative value of average equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We calculate the correction of e-folding number in the oscillating inflation stage, and then derive the viable parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The results indicate that the parameters are limited to γ < 5 × 10−4 and ζ > 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Moreover, r has an upper limit ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='006, but no lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' ACKNOWLEDGMENTS QYW and YT thank Shi Pi for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' YT is supported by National Key Re- search and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2021YFC2201901), and Natural Sci- ence Foundation of China (NSFC) under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 11851302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' YLW is supported by the Na- tional Key Research and Development Program of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2020YFC2201501, and NSFC under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 11690022, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 11747601, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 12147103, and the Strategic Prior- ity Research Program of the Chinese Academy of Sciences under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' XDB23030100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 20 Appendix A: Analytical treatment of Starobinsky inflation We give an analytical calculation of the tensor-to-scalar ratio r and spectral index ns in the Starobinsky inflationary model, namely, the Einstein gravity modified by a R2 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The effective scalar potential can be written as V (φ) = 1 8α � 1 − e−√ 2/3φ�2 , (A1) where α is the coefficient of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The relevant two slow-roll parameters are computed as ϵ = 4 3 1 � e √ 2/3φ − 1 �2, η = −4 3 e √ 2/3φ − 2 � e √ 2/3φ − 1 �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A2) Since inflation ends when ϵ ∼ 1 is reached first (η ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='15), we have φe = � 3 2 ln � 1 + 2 √ 3 � ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='94MP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A3) Then according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (23), the e-folding number is N = � 3 4 � e √ 2/3φ − � 2 3φ ��φe φi = 3 4 � e √ 2/3φi − e √ 2/3φe − � 2 3(φi − φe) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A4) For N ∼ (50, 60), we find that approximately φi ≃ � 3 2 ln �4 3(N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A5) Substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A2), we finally derive r = 16ϵ = 12 (N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55)2, (A6) ns = 1 − 6ϵ + 2η = 1 − 2 N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55 − 3 (N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A7) These results are shown as the red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Guth, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 23, 347-356 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Linde, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 108, 389-393 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Mukhanov, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Feldman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Brandenberger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 215, 203-333 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 21 [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Akrami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [Planck], Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 641, A10 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Mukhanov and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Chibisov, JETP Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 33, 532-535 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hawking, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 115, 295 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Guth and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Pi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 49, 1110-1113 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Starobinsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 117, 175-178 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Bardeen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Steinhardt and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Turner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 28, 679 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Weyl, Sitzungsber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Preuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Berlin (Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' ) 1918, 465 (1918).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Weyl, Annalen Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 59, 101-133 (1919).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Smolin, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 160, 253-268 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Cheng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 61, 2182 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [14] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Nishino and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rajpoot, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 79, 125025 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [15] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Romero, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Fonseca-Neto and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Pucheu, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 29, 155015 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [16] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Bars, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Steinhardt and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Turok, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 89, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4, 043515 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [17] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Quiros, [arXiv:1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2643 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [18] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Scholz, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 47, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2, 7 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [19] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ohanian, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3, 25 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ferreira, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hill and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ross, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 95, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4, 043507 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' de Cesare, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Moffat and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Sakellariadou, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C 77, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='9, 605 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [22] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ferreira, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hill and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ross, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 98, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='11, 116012 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ferreira, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hill, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Noller and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ross, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 97, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='12, 123516 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Tang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 784, 163-168 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [25] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ghilencea and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lee, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 99, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='11, 115007 (2019) [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wetterich, [arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='04741 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [27] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Tang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 803, 135320 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Tang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wu, JCAP 03, 067 (2020) [29] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ghilencea, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C 82, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1, 23 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [30] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ghilencea and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Harko, [arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='07056 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [31] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 93, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2, 024012 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [32] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wu, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C 78, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1, 28 (2018) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1140/epjc/s10052-017-5504-3 [arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='04537 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 22 [33] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wu, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='28, 2143001 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1142/S0217751X21430016 [arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='05404 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='gen-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [34] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wu, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='28, 2143002 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1142/S0217751X21430028 [arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='11078 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='gen-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [35] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ghilencea, JHEP 03, 049 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [36] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ferreira, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hill, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Noller and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ross, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 100, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='12, 123516 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ghilencea, JHEP 10, 209 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [38] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Oda, [arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='01437 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ghilencea, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C 80, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='12, 1147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [40] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Oda, PoS CORFU2019, 070 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [41] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Tang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 809, 135716 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [42] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Oda, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='37, 2050304 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [43] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ghilencea, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C 81, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='6, 510 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [44] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Cai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hao and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wang, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 74, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='9, 095401 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [45] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Tang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 106, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2, 023502 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Starobinsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 91, 99-102 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [47] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Vilenkin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 32, 2511 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Mijic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Morris and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Suen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 34, 2934 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [49] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Maeda, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 37, 858 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Aoki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Kubo and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Yang, JCAP 01, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='01, 005 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [51] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Stelle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 16, 953-969 (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [52] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Whitt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 145, 176-178 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [53] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Barrow and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Cotsakis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 214, 515-518 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [54] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [BICEP and Keck], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 127, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='15, 151301 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [55] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Huang, JCAP 02, 035 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [56] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Asaka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Iso, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Kawai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Kohri, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Noumi and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Terada, PTEP 2016, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='12, 123E01 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [57] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Cheong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lee and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Park, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 805, 135453 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [58] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rodrigues-da-Silva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Bezerra-Sobrinho and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Medeiros, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 105, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='6, 063504 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [59] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ivanov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ketov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Pozdeeva and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Vernov, JCAP 03, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='03, 058 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 23 [60] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Shtanov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Sahni and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Mishra, [arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='01828 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [61] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Modak, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' R¨over, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Sch¨afer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Schosser and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Plehn, [arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='05698 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='CO]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [62] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Damour and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Mukhanov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 80, 3440-3443 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [63] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Liddle and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Mazumdar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 58, 083508 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [64] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Taruya, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 59, 103505 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [65] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Cardenas and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Palma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 61, 027302 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [66] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Koh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Sin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lee, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 61, 027301 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [67] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Sahni and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 62, 103517 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [68] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Tsujikawa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 61, 083516 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [69] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Sami, Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Cosmol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 8, 309-312 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [70] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Dutta and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Scherrer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 78, 083512 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [71] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Johnson and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Kamionkowski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 78, 063010 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [72] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Mohseni Sadjadi and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Goodarzi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 732, 278-284 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [73] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Cembranos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Maroto and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' N´u˜nez Jare˜no, JHEP 03, 013 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [74] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Goodarzi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Mohseni Sadjadi, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C 77, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='7, 463 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [75] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [Planck], Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 641, A6 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} diff --git a/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf b/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e1fd5ae843ad685227d3bebd20ea07448a40db37 --- /dev/null +++ b/39FKT4oBgHgl3EQfRS0O/content/2301.11770v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eb08c8439ac709c9766b010874415a8f5d7e633fbbaedd33f2ff29cde7bf0d31 +size 205810 diff --git a/39FKT4oBgHgl3EQfRS0O/vector_store/index.faiss b/39FKT4oBgHgl3EQfRS0O/vector_store/index.faiss new file mode 100644 index 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b/3dFKT4oBgHgl3EQfQy0a/content/tmp_files/2301.11768v1.pdf.txt @@ -0,0 +1,5350 @@ +Prepared for submission to JHEP +FERMILAB-PUB-23-028-T, +IPPP/23/05 +Jet-veto resummation at N3LLp+NNLO in boson +production processes +John M. Campbell,a R. Keith Ellis,b Tobias Neumann,c Satyajit Sethd +aFermilab, PO Box 500, Batavia IL 60510-5011, USA +bInstitute for Particle Physics Phenomenology, Durham University, Durham, DH1 3LE, UK +cDepartment of Physics, Brookhaven National Laboratory, Upton, New York 11973, USA +dPhysical Research Laboratory, Navrangpura, Ahmedabad - 380009, India +E-mail: johnmc@fnal.gov, keith.ellis@durham.ac.uk, tneumann@bnl.gov, +seth@prl.res.in +Abstract: Vetoing energetic jet activity is a crucial tool for suppressing backgrounds and +enabling new physics searches at the LHC, but the introduction of a veto scale can introduce +large logarithms that may need to be resummed. We present an implementation of jet-veto +resummation for color-singlet processes at the level of N3LLp matched to fixed-order NNLO +predictions. Our public code MCFM allows for predictions of a single boson, such as Z/γ∗, +W ± or H, or with a pair of vector bosons, such as W +W −, W ±Z or ZZ. The implementation +relies on recent calculations of the soft and beam functions in the presence of a jet veto +over all rapidities, with jets defined using a sequential recombination algorithm with jet +radius R. However one of the ingredients that is required to reach full N3LL accuracy is only +known approximately, hence N3LLp. We describe in detail our formalism and compare with +previous public codes that operate at the level of NNLL. Our higher-order predictions improve +significantly upon NNLL calculations by reducing theoretical uncertainties. We demonstrate +this by comparing our predictions with ATLAS and CMS results. +arXiv:2301.11768v1 [hep-ph] 27 Jan 2023 + +Contents +1 +Introduction +1 +2 +Jet-veto factorization and resummation +2 +2.1 +The collinear anomaly coefficient and its approximations +5 +3 +Setup for phenomenology +9 +3.1 +Input parameters +9 +3.2 +Uncertainty estimates at fixed order +10 +3.3 +Uncertainty estimates at the resummed and matched level +11 +3.4 +Effects of cuts on rapidity at fixed order +13 +4 +Comparison with JetVHeto +15 +5 +Phenomenological results +17 +5.1 +Z and W production +17 +5.2 +W +W − production +21 +5.3 +W ±Z production +22 +5.4 +ZZ production +25 +5.5 +Higgs production +25 +6 +Conclusions +28 +A Reduced beam functions +30 +A.1 Structure of the two-loop reduced beam function +31 +B Definition of the beta function and anomalous dimensions +32 +B.1 +Expansion of β-function +32 +B.2 +Cusp Anomalous Dimension +34 +B.3 +Non-cusp anomalous dimension +35 +C Definitions for beam function ingredients +37 +C.1 Exponent h +37 +C.2 One loop splitting functions +38 +C.3 Two loop splitting functions +38 +C.4 P (1) ⊗ P (1) and R(1) ⊗ P (1) +41 +D Rapidity anomalous dimension +42 +D.1 dveto +2 +expansion +43 +E Renormalization Group Evolution +44 +– i – + +E.1 +Recovery of the double log formula +45 +F The hard function for the Drell-Yan process +46 +G The hard function for Higgs production +48 +G.1 Implementation of one-step procedure +48 +G.2 Implementation of the two-step procedure +50 +G.3 Assessment of the two schemes for the Higgs hard function +52 +1 +Introduction +Jet vetoing is a crucial technique in particle physics that is used primarily to suppress +backgrounds in processes involving the production of W +W − final states (e.g. directly or +in Higgs decays). By identifying and removing events that contain energetic hadronic jets +(vetoing), the impact of the dominant top-quark pair production background is reduced. +The concrete jet-veto implementation depends on factors such as the jet algorithm and its +parameters, as well as the kinematic selection cuts applied to the identified jets. For LHC +analyses, the most common jet vetoing scheme is to impose a maximum transverse momentum +cut pveto +T +on anti-kT jets. +The jet veto scale pveto +T +can induce large logarithms if it is smaller than the hard process scale +Q, which then mandates resummation. In this paper we describe a coherent implementation of +jet veto resummation in processes involving the production of a color-singlet boson (W, Z/γ∗ +and H bosons) or a pair of bosons (ZZ, W ±Z, and W +W −). Our resummation operates at +the level of N3LLp1 matched to fixed order NNLO. +We build on the pioneering work of previous studies, which have demonstrated the effectiveness +of resummation methods for a jet veto [1–5]. General purpose implementations include a +numerical approach to resummation at NNLO+NNLL [6, 7] and an automated approach to jet +veto studies at NLO+NNLL [8]. Publicly available codes operating at NNLL and addressing +the same issue are, JetVHeto [9], the code MCFM-RE [10] which is derivative of both MCFM +and JetVHeto, and MATRIX+RadISH [11]. Both JetVHeto and RadISH implement the same +analytic resummation formula of ref. [5]. +Our research extends and improves upon these earlier results through detailed phenomenological +studies of specific final states, including Higgs boson production [5, 12–14], W +W − production +[15, 16], and ZZ and W ±Z production [17]. Another important aspect of our study is the +performance of the resummation at N3LLp accuracy, which has not always been the case in +1The last missing ingredient for N3LL resummation is the exact dveto +3 +(the three-loop rapidity anomalous +dimension) which we approximate and take into account with an uncertainty estimate. We discuss this in detail +in the subsequent section. +– 1 – + +previous work. We also describe our approximation of the missing dveto +3 +that would be necessary +to reach full N3LL accuracy. Finally, we include our results in MCFM, a publicly distributed +code, which allows users to easily perform studies in practice. +Resummation of jet-veto logarithms has a close relationship with the resummation of transverse +momentum logarithms. In the latter, one is interested in transverse momenta all the way down +to zero pT , so the logarithms can be larger than in jet-veto processes where pveto +T +in the range +25 to 30 GeV is used experimentally. In this paper we explore which jet-veto processes actually +require resummation at these values of pveto +T +, supply the best predictions for those processes +where it is warranted, and confront our theoretical predictions with experimental data where +it is available. +In Section 2 we discuss the jet-veto factorization theorem including its ingredients that result +in the resummation. We describe our setup for phenomenology including our uncertainty +procedure in Section 3, compare with the public code JetVHeto in Section 4, and study the +phenomenological implications for a wide range of processes in Section 5. We conclude in +Section 6. +2 +Jet-veto factorization and resummation +We consider processes where jets have been defined using sequential recombination jet algorithms +[18] with distance measure +dij = min(k2n +Ti, k2n +Tj) +∆y2 +ij + ∆φ2 +ij +R2 +, +diB = k2n +Ti , +(2.1) +where the choice n = −1 is the anti-kT algorithm [19], n = 0 is the Cambridge-Aachen algorithm +[20, 21], and n = 1 is the kT algorithm [22, 23]. kTi denotes the transverse momentum of +(pseudo-)particle i with respect to the beam direction, and ∆yij and ∆φij are the rapidity and +azimuthal angle differences of (pseudo-)particles i and j. +To describe the resummation method we focus on the simplest case of quark-antiquark induced +Drell-Yan production of a lepton pair of invariant mass Q and rapidity y. The case of gluon +initiated processes is structurally the same, but with different ingredients that we give below +and in the appendices. In the presence of a jet veto over all rapidities we have a factorization +formula [3, 12, 13], +d2σ(pveto +T +) +dQ2dy += dσ0 +dQ2 +��CV (−Q2, µ) +��2 +× +� +Bq(ξ1, Q, pveto +T +, R, µ, ν) B¯q(ξ2, Q, pveto +T +, R, µ, ν) S(pveto +T +, R, µ, ν) +� ++ O +�pveto +T +Q +� +(2.2) +where ξ1,2 = (Q/√s) e±y and, +dσ0 +dQ2 = +4πα2 +3NcQ2s . +(2.3) +– 2 – + +Table 1: Counting of orders in the resummation, adapted from ref. [26]. The second column +indicates the nominal order when counting L⊥ ∼ 1/αs. +The third column states which +logarithms are included. The last three columns show the necessary additional anomalous +dimensions and hard function corrections in each successive order. The requisite anomalous +dimensions are provided in Appendix B. +Approximation +Nominal order +Accuracy ∼ αn +s Lk +⊥ +Γcusp +γcoll. +H +LL +α−1 +s +2n ≥ k ≥ n + 1 +Γ0 +tree +tree +NLL+LO +α0 +s +2n ≥ k ≥ n +Γ1, +γ0 +tree +N2LL+NLO +α1 +s +2n ≥ k ≥ max(n − 1, 0) +Γ2 +γ1 +1-loop +N3LL +NNLO +α2 +s +2n ≥ k ≥ max(n − 2, 0) +Γ3 +γ2 +2-loop +In this equation CV is a matching coefficient whose square is the hard coefficient function that +corrects the lowest order cross-section, see Eq. (2.3). Bq and B¯q are the quark beam functions +which describe the emission of radiation collinear to the two beam directions in the presence of +a jet veto, and S describes the emission of soft radiation in the presence of a jet veto. The +quantity ν is a supplementary scale necessitated by the rapidity divergences present in beam +and soft functions. The main process-independent ingredients are the beam and soft functions +for both incoming quarks and gluons which have been published recently at the two-loop level +[24, 25]. The hard function is process specific. We have used the existing two-loop fixed order +implementations in MCFM. +Overall the factorization theorem achieves a separation of scales. The hard function contains +logarithms of the ratio Q2/µ2, which can be minimized by setting µ2 = µ2 +h ∼ Q2. However, +inside the beam and soft functions, it is natural to choose µ = pveto +T +to avoid large logarithms. +The resummation of large logarithms is achieved by choosing µ ∼ Q in the hard function and +evolving it down to the resummation scale µ ∼ pveto +T +using the renormalization group (RG). +For the hard function the evolution is solved analytically, see Appendix E. +In RG-improved power counting the logarithms L⊥ = 2 log(µh/pveto +T +), where µh is of order Q, +are assumed to be of order 1/αs. With this definition the counting of powers of αs and of the +large logarithm L⊥ is shown in Table 1. The non-logarithmic terms that the resummation +does not provide are easily accounted for by adding the matching corrections. The matching +corrections are a finite contribution and add the effect of fixed-order corrections while removing +the logarithmic overlap through a fixed-order expansion of the resummation. +2.0.1 +Soft function +The jet veto soft function has been calculated using an exponential regulator [27] in Ref. [25]. +The calculation is divided into the sum of the soft function for a reference observable and a +correction factor, +S(pveto +T +, R, µ, ν) = S⊥(pveto +T +, µ, ν) + ∆S(pveto +T +, R, µ, ν) . +(2.4) +– 3 – + +In Ref. [25] the reference observable is the transverse momentum of the color singlet system +denoted by S⊥. S⊥ can be derived from the expression given in Refs. [28, 29] after performing +the Fourier transform to momentum space (see, for instance, the rules given in Table 1 of +Ref. [30]). ∆S depends on the jet algorithm and contributes for two or more emissions. It thus +depends only on double real emission diagrams. +2.0.2 +Refactorization and reduced beam functions +For consistency with the transverse momentum resummation framework in CuTe-MCFM [31] we +cast the factorization theorem in terms of the collinear anomaly framework. In this framework +the rapidity logarithms are exponentiated directly instead of resummed by solving rapidity RG +equations [32, 33]. For this we rewrite the square bracket in Eq. (2.2) as follows, +Bq(ξ1, Q, pveto +T +, R, µ, ν) B¯q(ξ2, Q, pveto +T +, R, µ, ν)S(pveto +T +, R, µ, ν) += +� Q +pveto +T +�−2Fqq(pveto +T +,R,µ) +e2hF (pveto +T +,µ) ¯Bq(ξ1, pveto +T +, R, µ) ¯B¯q(ξ2, pveto +T +, R, µ) . +(2.5) +The ν dependence vanishes in this combination of beam and soft functions. +We have factored out ehF/A(pveto +T +,µ) from each beam function, resulting in the reduced beam +functions ¯B. By construction hF/A are solutions of the RGE equation, +d +d ln µ hF/A(pveto +T +, µ) = 2ΓF/A +cusp(µ) ln +µ +pveto +T +− 2γq/g(µ) , +(2.6) +with boundary condition hF/A(µ, µ) = 0. The superscripts F or A signify whether the color +is treated in the fundamental (F) or adjoint (A) representation, corresponding to a quark +initiated process or a gluon initiated process, respectively. The exact form of hF/A(pveto +T +, µ), +determined by solving Eq. (2.6), is given in Appendix C.1. In terms of the reduced beam +functions the jet-vetoed cross-section is now given by, +d2σ(pveto +T +) +dQ2dy += dσ0 +dQ2 ¯H(Q, µ, pveto +T +) ¯Bq(ξ1, pveto +T +, R, µ) ¯B¯q(ξ2, pveto +T +, R, µ) + O(pveto +T +/Q) , +(2.7) +The choice of hF/A in Eq. (2.6) divides Eq. (2.2) into two separately RG invariant pieces, the +product of the two reduced beam functions ( ¯Bq ¯B¯q), and the hard function, ( ¯H) +¯H(Q, µ, pveto +T +) = +��CV (−Q2, µ) +��2 � Q +pveto +T +�−2Fqq(pveto +T +,R,µ) +e2hF (pveto +T +,µ) . +(2.8) +For quark-initiated processes the functions CV and Fqq obey the following RG equations. +d +d ln µ CV (−Q2, µ) = +� +ΓF +cusp(µ) ln −Q2 +µ2 ++ 2γq(µ) +� +CV (−Q2, µ) , +(2.9) +d +d ln µFqq(pveto +T +, R, µ) = 2ΓF +cusp(µ) . +(2.10) +– 4 – + +Eqs. (2.9) and (2.10) are structurally the same for the gluon case with different anomalous +dimensions. +The function ¯H is RG invariant due to the RGE’s satisfied by CV and Fqq and hF : +d +dµ +¯H(Q, µ, pveto +T +) = O(α3 +s) . +Consequently, the remaining product of reduced beam functions is also RG invariant, up to +the order calculated. In our case, +d +dµ +¯Bq(ξ1, pveto +T +, R, µ) ¯B¯q(ξ2, pveto +T +, R, µ) = O(α3 +s) . +(2.11) +The confirmation of Eq. (2.11) and the confirmation of the R-dependence of the collinear +anomaly given in the next section are two simple checks of the results of Refs. [24, 25]. Full +details of the formulas needed to perform this check are given in Appendix C. +If the scale pveto +T +is in the perturbative domain, the reduced beam function can be written in +terms of the matching kernels ¯I as +¯Bi(ξ, pveto +T +, R, µ) = +� +j=g,q,¯q +� 1 +ξ +dz +z +¯Iij(z, pveto +T +, R, µ) φj/P (ξ/z, µ) , +where φ denotes the usual collinear parton distribution of a parton of flavor j in a proton P. +The matching coefficients ¯I are extracted from I, the two-loop beam and soft functions of +Refs. [24, 25] as, +¯Iij(z, pveto +T +, R, µ) = e−hF/A(pveto +T +,µ) Iij(z, pveto +T +, R, µ) . +(2.12) +The coefficients in Ref. [24] are presented as a Laurent expansion in the jet radius parameter +R. Analytic expressions are presented for all flavor channels except for a set of R-independent +non-logarithmic terms which are presented as numerical grids. For our purposes we have +interpolated the numerical grids using a spline fit. We give further details on the reduced beam +functions in Appendix A. +2.1 +The collinear anomaly coefficient and its approximations +The missing ingredient for a complete N3LL resummation is the three-loop collinear anomaly +coefficient and therefore warrants a longer discussion. This limitation has been discussed in the +literature and approximated in various ways. Here we discuss the uncertainty associated with +the approximations and how we take it into account in our phenomenological predictions. +As presented in Eq. (2.10) the collinear anomaly coefficients obey the RG equations, +d +d ln µFqq(pveto +T +, R, µ) = 2ΓF +cusp(µ) , +(2.13) +d +d ln µFgg(pveto +T +, R, µ) = 2ΓA +cusp(µ) , +(2.14) +– 5 – + +where, for example, Fqq has the expansion, +Fqq(pveto +T +, R, µ) = αs +4πF (0) +qq (pveto +T +, R, µ) + +�αs +4π +�2 +F (1) +qq (pveto +T +, R, µ) ++ +�αs +4π +�3 +F (2) +qq (pveto +T +, R, µ) + +�αs +4π +�4 +F (3) +qq (pveto +T +, R, µ) + . . . +(2.15) +While the logarithmic structure is given by the RG equations, the constant boundary parts +dveto +k +(R, B) where B = F or A need to be determined by separate calculations and are also +referred to as the rapidity anomalous dimensions in the framework of Refs. [32, 33]: +F (0) +qq (pveto +T +, R, µh) = ΓF +0 L⊥ + dveto +1 +(R, F) , +F (1) +qq (pveto +T +, R, µh) = 1 +2ΓF +0 β0L2 +⊥ + ΓF +1 L⊥ + dveto +2 +(R, F) , +F (2) +qq (pveto +T +, R, µh) = 1 +3ΓF +0 β2 +0L3 +⊥ + 1 +2(ΓF +0 β1 + 2ΓF +1 β0)L2 +⊥ ++ (ΓF +2 + 2β0dveto +2 +(R, F))L⊥ + dveto +3 +(R, F) , +F (3) +qq (pveto +T +, R, µh) = 1 +4β3 +0ΓF +0 L4 +⊥ + (ΓF +1 β2 +0 + 5 +6ΓF +0 β0β1)L3 +⊥ ++ (1 +2ΓF +0 β2 + ΓF +1 β1 + 3 +2ΓF +2 β0 + 3dveto +2 +(R, F)β2 +0)L2 +⊥ ++ (ΓF +3 + 3dveto +3 +(R, F)β0 + 2dveto +2 +(R, F)β1)L⊥ + dveto +4 +(R, F) . +(2.16) +The analogous expression for gluons (F → A) is given in Eq. (D.1). The coefficients in the +expansion of the cusp anomalous dimension, ΓF +k , are given in Appendix B.2. +For single gluon emission dveto +1 +(R, B) = 0. The function dveto +2 +is defined below in Eq. (2.17). +There is only partial information on dveto +3 +from Refs. [14, 34, 35], and we have to rely on +an approximation. +To estimate the validity of this approximation we first study similar +approximations of dveto +2 +. +The function dveto +2 +is given by [12], +dveto +2 +(R, B) = dB +2 − 32CB f(R, B) , where +dB +2 = CB +��808 +27 − 28ζ3 +� +CA − 224 +27 TF nf +� +. +(2.17) +The function f(R, B), which gives the dependence on the jet radius R, is known as an expansion +about R = 0 up to terms including R4, +f(R, B) = CB +� +− π2R2 +12 ++ R4 +16 +� ++ CA +� +cA +L ln R + cA +0 + cA +2 R2 + cA +4 R4 + . . . +� ++ TF nf +� +cf +L ln R + cf +0 + cf +2R2 + cf +4R4 + . . . +� +. +(2.18) +The terms on the first line are due to independent emission, whereas the terms on the second +and third lines are due to correlated emission [4]. The expansion coefficients are given in +Appendix D in analytic and numerical form. +– 6 – + +2.1.1 +Approximations for dveto +2 +Using Eqs. (2.17) and (D.4) we have for the gluon case in the limit nf → 0 and retaining only +logarithmic and constant terms in R, +dveto +2 +(R, A) = −32C2 +A +� +− +1 +32C2 +A +dA +2 + cA +L ln R + cA +0 +� +≃ −32C2 +A +� +− 1.096259 ln R + 0.7272641] +∼ 32C2 +A × ln +�R +2 +� +. +(2.19) +This result was used as a basis for an approximation to dveto +3 +in ref. [12]. However, the leading +color (nf = 0) approximation is rather poor. With full nf dependence, but retaining only +logarithmic and constant terms in R and setting nf = 5 we have +dveto +2 +(R, B) = 32CBCA +� +(1.096 + 0.0295nf) ln R − (0.72726 + 0.12445nf) +� +∼ 32CBCA +� +1.2435 ln +� R +2.96 +�� +. +(2.20) +In Fig. 1 we show dveto +2 +(R, A) and its approximations in units of dA +2 as a function of the jet +radius R. As a reminder, dA +2 is the non-R dependent part of d2, see Eq. (2.17). We first +compare the full result (red) with the inclusion of terms up order R2 (green). This shows +that the R expansion converges quickly and it is sufficient to consider only terms up to R4 for +practical applications. Including only the logarithm and the constant (blue) gives a reasonable +approximation for sufficiently small R, with percent-level deviations around R = 0.4. The +leading color approximation (magenta) works only crudely as a first guess and could be used +in the absence of any better estimate. +2.1.2 +The function dveto +3 +While the complete dveto +3 +is unknown so far, we can extract the leading logarithmic term +from results in the literature. Given that this approximation works reasonably well for dveto +2 +for R ∼ 0.4, it is reasonable to expect a similar behavior for dveto +3 +. We further estimate the +uncertainty associated with such an approximation. +From Eq. (2.15) the collinear anomaly coefficient at µ = pveto +T +is given by, +Fgg(pveto +T +, R, pveto +T +) = +�αs +4π +�2 +dveto +2 +(R, A) + +�αs +4π +�3 +dveto +3 +(R, A) + . . . +(2.21) +Therefore, expanding the collinear anomaly we have that +� Q +pveto +T +�−2Fgg(pveto +T +,pveto +T +) +=1 − 2 +�αs(pveto +T +) +4π +�2 +ln +� Q +pveto +T +� +dveto +2 +(R, A) +− 2 ln +�αs(pveto +T +) +4π +�3 +ln +� Q +pveto +T +� +dveto +3 +(R, A) + O(α4 +s). +(2.22) +– 7 – + +Figure 1: Approximations of dveto +2 +(R, A) +scaled by the constant dA +2 . The full result, +Eq. (2.17) is plotted in red. The approxi- +mation retaining only constant terms and +logarithms of R is shown in blue. The ap- +proximation retaining constant terms and +logarithms of R and R2 terms is shown in +green. The leading color ansatz, Eq. (2.19), +derived setting nf = 0, is 32C2 +A ln(R/2) and +is shown in magenta. The red, blue and green +curves are all plotted for nf = 5. +Figure 2: Effect of R0 variation in dveto +3 +as +given by Eq. (2.24) with nf = 5, compared to +the case dveto +3 += 0: R0 = 1 (black), R0 = 0.5 +(red, dashed), R0 = 2 (blue, dashed). +At order α3 +s the leading term in the limit R → 0 can be extracted from Eq. (C.2) of Ref. [14] +which reads, +Fcorrel +LLR,31(R) = +�αs +4π +�3 +ln +� Q +pveto +T +� +· 128CA ln2 R +R0 +× +� +1.803136C2 +A − 0.589237nf2TRCA + 0.36982CF nf2TR − 0.05893n2 +f4T 2 +R +� +. +(2.23) +Comparing the third-order coefficient in the two equations we thus have for a general color +representation +dveto +3 +(R, B) = −64CB ln2 � R +R0 +� +(1.803136C2 +A + 0.36982CF nf − 0.589237CAnf − 0.05893n2 +f) += −8.38188 × 64CB ln2 � R +R0 +� +for nf = 5 . +(2.24) +Hence, the sign of the leading term in the small R limit is known. In this limit dveto +3 +leads to an +increase in the cross-section. This approximation only gives the leading R behavior, and it has +been suggested that one may plausibly take 1 +2 < R0 < 2 as an uncertainty envelope [14]. +Since dveto +3 +enters through the collinear anomaly as an overall factor, we consider the impact of +varying R0 in Fig. 2. For typical values of pveto +T += 30 GeV (as considered in this paper for the +– 8 – + +Table 2: Input and derived parameters used for our numerical estimates. +MW +80.385 GeV +ΓW +2.0854 GeV +MZ +91.1876 GeV +ΓZ +2.4952 GeV +Gµ +1.166390 × 10−5 GeV−2 +mt +173.2 GeV +mh +125 GeV +m2 +W = M2 +W − iMW ΓW +(6461.748225 − 167.634879 i) GeV2 +m2 +Z = M2 +Z − iMZΓZ +(8315.17839376 − 227.53129952 i) GeV2 +cos2 θW = m2 +W /m2 +Z +(0.7770725897054007 + 0.001103218322282256 i) +α = +√ +2Gµ +π +M2 +W (1 − M2 +W +M2 +Z ) +7.56246890198475 × 10−3 giving 1/α ≈ 132.23 . . . +comparison with experimental studies) there is an effect of less than two percent for R = 0.4. +This is in agreement with the deviations we found for dveto +2 +for this approximation. +We take into account this variation in our uncertainty estimates, see Section 3.3. A definitive +statement on this issue will have to await an exact calculation of dveto +3 +. +3 +Setup for phenomenology +Before discussing phenomenological results, we list our input parameters, the method for +matching to fixed order, and the approach for estimating uncertainties at fixed order and at +the resummed level. +3.1 +Input parameters +The input values used in our numerical studies are shown in Table 2. +As indicated in +the table we use the complex mass scheme for the W and Z boson masses. The number +of light quarks, nf, is set equal to five, except for the case of W +W −-production where +nf = 4. We use the PDF distribution NNPDF31_nnlo_as_0118 except for W +W − where we use +NNPDF31_nnlo_as_0118_nf_4 [36]. Note that we use these NNLO parton distributions even in +our lower order predictions. +In the cases of WW and ZZ production, at O(α2 +s) the cross-section receives contributions from +processes with two gluons in the initial state. When performing the resummed calculations we +only include such contributions at NLL. However, these contributions only represent about 3% +of the cross-section for pveto +T += 10 GeV, rising to about 6–8% for pveto +T += 60 GeV. Therefore, +neglecting higher order corrections to these contributions, which are not implemented in our +code, is justified. Although only strictly true for the leading q¯q component we refer to the full +resummed calculation as N3LLp. +We match the resummation and fixed-order NkLO corrections using a naive additive scheme as +– 9 – + +follows, +σN(k+1)LL+N(k)LO(pveto +T +) = σN(k+1)LL(pveto +T +) + σ∆,k(pveto +T +) , where +(3.1) +σ∆(pveto +T +) = σNkLO(pveto +T +) − dσN(k+1)LL(pveto +T +) +���� +exp. to NkLO +. +(3.2) +The matching correction σ∆(pveto +T +) is defined as a function of pveto +T +, using the difference between +the fixed-order contribution and the resummed result expanded to the same fixed order. The +limit pveto +T +→ 0 of σ∆(pveto +T +) is finite, which also allows its use as a higher-order subtraction +scheme. +The use of a naive matching without a transition mechanism that switches off the resummation +at large pveto +T +is justified since the matching corrections for all considered cases in this paper are +small; even in the most extreme case they are less than 20%. In other words, the resummation +alone provides a good description of the cross-sections and does not need to be switched off. +Any transition function to turn off the resummation at large pveto +T +would have a very small +effect. This is in contrast to transverse-momentum resummation where a transition function is +necessary [31]. +3.2 +Uncertainty estimates at fixed order +Ultimately the resummed predictions should offer a practical advantage compared to the +fixed-order predictions. In many cases, the quantity log(Q/pveto +T +) is not very large, and it may +not seem worthwhile to use resummed results. However, as we will show, the resummation +works remarkably well on its own and has matching corrections of only up to around 20%, often +much less. The clear separation of scales and the resummation then allow for smaller and more +reliable uncertainty estimates. To set the stage, we first examine perturbative convergence and +uncertainties at fixed order for quark and gluon induced boson processes, as well as for WW +and ZZ production. +Constructing jet-vetoed cross-sections at fixed order requires the combination of different +cross-sections. However, if we naively subtract the jet cross-section from the inclusive result, it +can result in underestimated uncertainties and narrowing uncertainty bands. To avoid this, +different methods have been proposed in the literature, of which we compare the following +two. +One strategy, which we term the "two-scale" approach, is to consider the different relevant +scales Q and pveto +T +of the vetoed cross-section σ0, and include both of them in the uncertainty +estimate through a multi-point variation around both scales [8]. To compute this uncertainty, +we separately vary the renormalization scale µr and the factorization scale µf over the values +{µh, 2µh, µh/2, pveto +T +, 2pveto +T +, pveto +T +/2}, where µh depends on the process under consideration. An +estimate of the uncertainty is then obtained by adding in quadrature the maximum deviations +from µr = µf = µh, from µr and µf variation separately. +– 10 – + +Another approach, advocated by Refs. [14, 37], takes the jet-veto efficiency (JVE) as the central +quantity, which is the ratio of jet-vetoed cross-section to total cross-section. By combining the +uncertainties of these two quantities in quadrature, one obtains a more robust estimate of the +uncertainty in the jet-vetoed cross-section. This is because the uncertainties are considered +uncorrelated: the uncertainties in the jet-veto efficiency are typically due to non-cancellation +of real and virtual contributions, while those in the total cross-section are connected with large +corrections from higher orders [14]. +For our JVE approach, we follow the simplest formulation (“scheme (a)” of Ref. [14]) to compute +a JVE-based uncertainty. For this we consider variation over the scales {µh, 2µh, µh/2} of σincl +and combine in quadrature the uncertainty from the calculation of the 0-jet efficiency (σ0/σincl) +and the uncertainty from the inclusive calculation. Our final fixed-order uncertainty band is +the envelope of the two-scale and JVE approaches. +With these procedures, our fixed-order results for Z and H production are shown in Fig. 3. +For Z production we use the canonical choice µh = Q, where Q is the invariant mass in the +final state. For Higgs production we use µh = Q/2, guided by the calculation of the inclusive +cross-section where such a choice results in markedly-improved perturbative convergence. We +observe that for Z production the NNLO uncertainty band is wholly contained within the NLO +one, while for the Higgs case the bands at least overlap somewhat throughout the range. For +Higgs production following the combined two-scale and JVE approach results in a significantly +larger uncertainty at both NLO and NNLO, especially at smaller values of pveto +T +. On the other +hand, for Z production the additional uncertainty from the JVE approach is very small and +negligible at NNLO. +Predictions for WW and ZZ production (with µh = Q) are shown in Fig. 4. The limited +overlap between the NLO and NNLO bands indicates that uncertainties are underestimated, +even with the generous scale uncertainty procedure that we follow. The additional uncertainty +resulting from the JVE procedure is small, especially at NNLO, because the scale uncertainty +of the inclusive cross-sections is very small. +3.3 +Uncertainty estimates at the resummed and matched level +For our central predictions, we set the resummation and factorization scales to µ = pveto +T +and +the hard scale (corresponding to the renormalization scale) to µh = Q, where Q is the invariant +mass of the color-singlet final state. The exception is Higgs production, where we choose +µh = Q/2 as previously discussed. For the collinear anomaly coefficient dveto +3 +, we use the form +given in Eq. (2.24) [14] with R0 = 1. +Complications arising at fixed order, described in Section 3.2, are not present in the resummed +case and therefore we can follow a simpler approach where we vary all scales in our formalism +and take the envelope, as detailed below. While the matching of resummed predictions to +fixed-order could still introduce a complication, the matching corrections are not dominant. +The bulk of the cross-section comes from the resummation and it allows us to follow the simple +– 11 – + +(a) Z production using the setup of ref. [38]. +(b) H production. +Figure 3: Comparison of NLO and NNLO fixed order predictions as a function of the jet veto. +Central predictions solid, uncertainty estimates using either the two-scale approach (dotted) +or the envelope of that and the JVE approach (dashed). +(a) WW production using the setup of ref. [39]. +(b) ZZ production using the setup of ref. [40]. +Figure 4: Comparison of NLO and NNLO fixed order predictions as a function of the jet veto. +Central predictions solid, uncertainty estimates using either the two-scale approach (dotted) +or the envelope of that and the JVE approach (dashed). +– 12 – + +procedure of varying all scales in the naively obtained (without JVE) jet-veto cross-section +too. +The small and narrowing uncertainty bands at fixed order would typically appear in regions +where the resummation is found to be dominant, i.e. where fixed-order contributes very little +through the matching corrections. In practice we observe that the size of uncertainties are +overall uniform in both the resummation and large pveto +T +fixed-order regions, as can be seen in all +of our following predictions. This supports the conclusion that our procedure is sufficient. +Overall, our procedure for estimating uncertainties is as follows. +1. For the resummation (fixed-order) parts we vary both the resummation (factorization) +and hard (renormalization) scales by a factor of two about their central values, adding +the excursions in quadrature to obtain the total scale uncertainty. +2. For the resummation we re-introduce the rapidity scale in Eq. (2.5) by re-writing the +collinear anomaly factor as follows [12, 41]: +� Q +pveto +T +�−2Fii(pveto +T +,R,µ) += +�Q +ν +�−2Fii(pveto +T +,R,µ)� ν +pveto +T +�−2Fii(pveto +T +,R,µ) +. +(3.3) +For ν ∼ pveto +T +the second factor can be expanded since it does not contain a large logarithm. +We vary the rapidity scale ν in the range [pveto +T +/2, 2pveto +T +] for gluon-initiated processes +and in the range [pveto +T +/6, 6pveto +T +] for quark-initiated processes. The large variation for +quark-initiated processes ensures overlapping uncertainty bands at NNLL and N3LLp; +this is achieved by the range given above, as demonstrated explicitly in Sections 4 and 5. +3. The parameter R0 in dveto +3 +is varied between 0.5 and 2. +We first combine the scale uncertainties (1 and 2) in quadrature and then, to obtain our total +uncertainty, add the variation of R0 (3) linearly. +3.4 +Effects of cuts on rapidity at fixed order +The usual jet veto resummation described so far imposes no cut on the jet rapidity. This is in +contrast to experimental analyses, see Table 3, which impose such a cut because of limited +detector acceptance and to diminish the effect of pileup. Ref. [42] identifies three different +regimes, depending on pt, Q and ycut. +• For pveto +T +/Q ≫ exp(−ycut) standard jet veto resummation should apply, effects due to +the rapidity cut are corrections power suppressed by Q exp(−ycut)/pveto +T +. +• For pveto +T +/Q ∼ exp(−ycut) the effects of a rapidity cut must be treated as a leading power +correction. +• For pveto +T +/Q ≪ exp(−ycut) the logarithmic structure is changed already at leading log +level, and non-global logarithms appear. +– 13 – + +Table 3: Jet rapidity cuts applied in the experimental studies examined later in this paper. +Process +Ref. +ycut +Higgs +– +no study +Z (CMS) +[38] +2.4 +W (ATLAS) +[43] +4.4 +WW (CMS) +[39] +4.5 +WZ (ATLAS) +[44] +4.5 +WZ (CMS) +[45] +2.5 +ZZ (CMS) +– +no study +(a) Z production following the setup of ref. [38]. +(b) H production. +Figure 5: Effect of the jet rapidity cut at NNLO with pveto +T += 30 GeV. +We estimate the practical impact of experimentally used jet rapidity cuts at fixed order. +Including the rapidity cut in the resummation requires large changes and ingredients, which +are also only available a low order so far [42]. +The effect of the jet rapidity cut for the Z and Higgs production cases is illustrated in Fig. 5. +These calculations are performed at NNLO for pveto +T += 30 GeV. The rapidity cut plays a bigger +role for Higgs production: for example for ycut = 2.5 the cross-section is 11% larger than +the result with no rapidity cut, compared to only 2% for Z production. This is due to the +larger logarithm (log(mH/pveto +T +)/ log(mZ/pveto +T +) ≈ 1.28) and the larger color prefactor (CA/CF += 2.25) in Higgs production. However, for ycut = 4.5 the effect of the rapidity cut is negligible +in both cases. +– 14 – + +(a) WW production. +(b) ZZ production. +Figure 6: Effect of the jet rapidity cut at NNLO with pveto +T += 30 GeV. +The corresponding results for diboson processes are shown in Fig. 6. In this case, the disparity +between Q and pveto +T +is much larger, so the rapidity cut can play a crucial role, although the +effect is still not as important as for Higgs production. For ycut = 2.5 the WW and ZZ +cross-sections 4% larger than the results with no rapidity cut, and the effect of ycut = 4.5 is +negligible. +4 +Comparison with JetVHeto +While jet-veto resummed phenomenology has been extensively studied in the literature, the +only public codes that permit detailed predictions use JetVHeto or RadISH. +For jet-veto +resummation RadISH implements the analytic JetVHeto resummation formula [5]. The codes +rely on the formalism of the CAESAR approach [4, 46] extended to NNLL [5]. An extension +of the RadISH code has been used to perform joint jet-veto and boson transverse momentum +resummation [47]. +For our comparisons we use RadISH version 3.0.0 [48, 49] and JetVHeto version 3.0.0 [5, 14, 37] +including small-R resummation [4, 35] as part of MCFM-RE [16]. Both codes operate at the +level of NNLL and we have checked that they give indeed the same results. +In our comparison, we would like focus on the differences in the resummation part, since +the fixed-order part is identical in each calculation. +We explore how central values and +uncertainties compare at NNLL to our results and in how far N3LLp results improve the +perturbative convergence. However, the matching to fixed-order is handled differently in each +formalism. Different matching schemes (e.g. additive or multiplicative schemes of various +– 15 – + +types) probe higher-order effects. It has also been advocated to match at the level of jet-veto +efficiencies [14]. Fortunately, matching corrections are generally small for jet-veto scales of +30 to 40 GeV for all considered boson and di-boson processes. We therefore focus on the +resummation in our comparison. +The JetVHeto formalism considers three scales µR, µF and Q that are all similar in magnitude +to the hard scale. To ensure that the resummation switches off for pveto +T +≳ Q, the resummed +logarithms are modified through the prescription log(Q/pveto +T +) → 1/p log((Q/pveto +T +)p + 1). For +JetVHeto p has a default value of 5 [14], while for RadISH the default choice is 4. For comparison +purposes we use p = 5 in both cases. It is evident that for sufficiently small pveto +T +the precise +value of p does not matter. Changing this parameter has a similar effect to turning off the +resummation with a transition function. In principle this demands a fully matched calculation, +but the matching corrections of our considered cases are small and we have checked that the +effect of changing p to 3 or 4 is subleading compared to the scale uncertainties. Here we focus +on those scale uncertainties. +In ref. [14] it has been argued that the Q should be varied by a factor of 3 +2 around its central +value, based on new insights from convergence at N3LO for Higgs production. For simplicity, +we use a more conservative variation by a factor of two. We independently vary µR, µF and Q +by a factor of two around a central scale of mℓℓ for Z-boson production and around mH/2 +for Higgs production. Our uncertainty bands for this comparison are obtained by taking the +envelope of these results. +Z-boson production +For the comparison of Z production we choose a central hard scale of mℓℓ with results shown +in Fig. 7. We find that our MCFM NNLL central values have only marginal compatibility with +our JetVHeto uncertainty estimates, despite having the same logarithmic order. This indicates +that the JetVHeto uncertainties (as estimated according to our procedure just described) do +not fully account for the higher-order corrections. On the other hand, our uncertainties at +NNLL are larger, leading to an overall agreement between the two methods. +At N3LLp uncertainties decrease dramatically compared to NNLL, but they are quite asymmetric, +which suggests that a symmetrization of uncertainties may be necessary in this case. We also +observe that without the large uncertainties at NNLL, there would be no overlap between the +N3LLp results and NNLL. This highlights the importance of carefully estimating and comparing +uncertainties to accurately assess the compatibility of different methods and results. +H-boson production +In our study of Higgs production, we choose a central hard scale of mH/2 and show results in +Fig. 8. All results are computed in the mt → ∞ theory and rescaled by a factor of 1.0653 to +account for finite top-quark mass effects, see Eq. (G.5). +– 16 – + +qq → Z → e+e−, s = 13 TeV, µh = me+e− +200 +400 +600 +800 +10 +20 +30 +40 +pt +veto [GeV] +σveto [pb] +RadISH/JetVHeto/MCFM−RE NNLL +MCFM NNLL +MCFM N3LLp +Figure 7: Comparison of JetVHeto NNLL resummation with our NNLL and N3LLp results for +Z production with cuts as in Table 4. +The Higgs case is distinct from Z production since it is gluon-gluon initiated instead of +quark-initiated. In this case, our predictions agree well with the JetVHeto results, but our +uncertainties at NNLL are again much larger. +Note that we vary the JetVHeto scale Q by a factor of two, while the JetVHeto authors vary by +a factor of 3/2 in the Higgs case. This difference in the amount of variation may require some +tuning in our formalism, at least at the NNLL level. However, the perturbative convergence is +again excellent with small uncertainties at N3LLp and central predictions that agree well with +NNLL. +5 +Phenomenological results +In this section, we present the results of our phenomenological studies, which are based +on the uncertainty procedure, matching to fixed-order, and input parameters described in +Section 3. We compare our findings with experimental results from the literature and discuss +their implications. +5.1 +Z and W production +The process of Z production has already been extensively studied in the literature, thus +enabling a variety of cross-checks of our calculation. The implementation of the hard function +and its evolution has been verified by comparison with the explicit results given in Table 1 of +ref. [50]. The full machinery of the resummation and matching procedure can also be compared +with the results of ref. [5], with which we find excellent agreement within uncertainties, see +also Section 4. +– 17 – + +gg → H, s = 13.6 TeV, µh = mH 2 +20 +30 +40 +20 +30 +40 +50 +pt +veto [GeV] +σveto [pb] +RadISH/JetVHeto/MCFM−RE NNLL +MCFM NNLL +MCFM N3LLp +Figure 8: Comparison of JetVHeto NNLL resummation with our NNLL and N3LLp results for +non-decaying H production. +Table 4: Cuts used in the analysis of Z production, adapted from ref. [38]. +lepton cuts +ql1 +T > 30 GeV, ql2 +T > 20 GeV, |ηl| < 2.4 +lepton pair mass +71 GeV < ml−l+ < 111 GeV +jet veto +anti-kT , R = 0.4, 0-jet events only +We first investigate the impact of choosing a time-like hard scale in the resummed result for +Z production. Previous work has shown that choosing a space-like hard scale (µ2 +h = Q2) +can lead to significant corrections in the perturbative expansion of some processes, while a +time-like hard scale (µ2 +h = −Q2) can resum certain π2 contributions [51] using a complex +strong coupling. +For this comparison we consider purely resummed results at NNLL and N3LLp, only considering +uncertainties originating from scale variation (items 1 and 2 of our uncertainty procedure in +Section 3.3). We consider the process pp → Z/γ∗ → ℓ−ℓ+, i.e. a final state of definite lepton +flavor. We use the same set of cuts and vetoes as in the √s = 13 TeV CMS analysis [38], but +extend the veto to jets of all rapidities, rather than only those with |y| < 2.4. This difference, +and the effect of matching to NNLO, is discussed in detail in Section 5.1.1. +Our results are shown in Fig. 9a as a function of the value of the jet veto. We observe that +the results do not depend strongly on the choice of hard scale, with a difference of about 4% +at NNLL and only 1% at N3LLp. This indicates that resumming the π2 terms results in only +– 18 – + +(a) Predictions are computed using a central choice +for the hard scale given by either µ2 +h = Q2 or +µ2 +h = −Q2. The lower panel shows the ratio of +the result for µ2 +h = −Q2 to the one for µ2 +h = Q2. +(b) Predictions and CMS measurement as ratio to +matched result. +Figure 9: Comparison of NNLL and N3LLp predictions for Z production as a function of the +jet veto, using the setup of ref. [38] (central predictions solid, uncertainty estimate according +to the text, dashed). +a small enhancement of the cross-section for W and Z production. Based on these findings, +we use the space-like hard scale (µ2 +h = Q2) in our subsequent studies of Z and W boson +production, as it is the more commonly used choice in the literature. +5.1.1 +CMS Z production +As previously mentioned, the CMS measurement we are comparing to includes a jet rapidity cut +of |y| < 2.4. To assess the importance of this restriction, we first compare the NNLO predictions +with and without the rapidity cut, as a function of the jet veto value. This comparison, shown +in Table 5, helps us better understand the limitations of our analysis. +We use the quantity ϵ(pveto +T +) to quantify the increase in the cross-section when the rapidity cut +is applied, defined as +ϵ(pveto +T +) = σ0−jet(ycut = 2.4) +σ0−jet(no ycut) +− 1 . +(5.1) +The experimental measurement we are comparing to uses a jet veto of pveto +T += 30 GeV, for which +the rapidity cut has only a 3% effect on the cross-section. This suggests that our calculation +with an all-rapidity jet veto is appropriate for comparing to the experimental measurement. +However, as pveto +T +decreases, the impact of the rapidity cut becomes more significant, until at +– 19 – + +Table 5: The Z + 0-jet cross-section prediction at NNLO (µ = Q), with and without a jet +rapidity cut. +pveto +T +[GeV] +5 +10 +20 +30 +40 +σ0−jet(no ycut) [pb] +140 +347 +539 +627 +675 +σ0−jet(ycut = 2.4) [pb] +242 +411 +569 +643 +685 +ϵ +0.73 +0.18 +0.06 +0.03 +0.01 +qq → Z → l+l−, s = 13 TeV, CMS cuts, arXiv:2205.02872 +618 ± 17 pb +592−13 ++9 pb +600 +650 +700 +750 +800 +CMS +NLO +NNLL +NNLL+NLO +NNLO +N3LLp +N3LL+NNLO +σveto [pb] +Figure 10: Comparison of Z-boson jet-vetoed predictions with the CMS [38] 13 TeV measure- +ment. Shown are results at fixed-order, purely resummed and matched. +pveto +T += 5 GeV it is no longer appropriate to neglect the rapidity cut. This is consistent with the +arguments of Ref. [42], which suggest that the standard jet veto resummation formalism should +suffice as long as ln(Q/pveto +T +) ≪ ycut. In our case, ln(Q/pveto +T +) ranges from 0.8 to 2.9 for pveto +T +from 40 down to 5 GeV, so the standard jet veto resummation should be appropriate, albeit +with sizeable power corrections, for ycut = 2.4 except for the smallest values of pveto +T +. +We now turn to a comparison with the CMS result [38], which uses a jet threshold of 30 GeV. +Our comparison with fixed-order, purely resummed and matched predictions is shown in Fig. 10. +We find that the fixed-order and resummed results differ by only a few percent, indicating +that resummation is not necessary for this value of the jet veto. This is because the quantity +ln(MZ/pveto +T +) = 1.1 is not large enough to require resummation. The CMS measurement yields +a cross-section of 618 ± 17 pb, while our best prediction is 592+9 +−13 pb. +We study the production of Z bosons as a function of the jet veto in Fig. 9b. We observe +that the difference between the resummed and central fixed-order results is small, even for the +smallest values of pveto +T +considered. However, the uncertainties in the fixed-order prediction are +larger across the whole range, particularly for small pveto +T +. For values of pveto +T +in the range of 20 +to 40 GeV, which are of practical interest, the N3LLp uncertainty is smaller than the NNLO +uncertainty by about a factor of 1.5. +5.1.2 +ATLAS W production +We now perform a comparison with √s = 8 TeV ATLAS data on W production [43]. For +this study, jets were identified using the anti-kT algorithm with R = 0.4 and must satisfy +– 20 – + +qq' → W± → eν, s = 8 TeV, ATLAS cuts, arXiv:1711.03296 + 4.72 ± 0.3 nb +4.71−0.1 ++0.07 nb +4.0 +4.5 +5.0 +5.5 +6.0 +ATLAS +NLO +NNLL +NNLL+NLO +NNLO +N3LLp +N3LL+NNLO +σveto [nb] +Figure 11: Comparison of W-boson jet-vetoed predictions with the ATLAS [43] 8 TeV mea- +surement. Shown are results at fixed-order, purely resummed and matched. +pT > 30 GeV and |y| < 4.4. We have checked at fixed order that this large rapidity cut has a +negligible impact of a few per mille, i.e. results are unchanged within the numerical precision +to which we work. +Summing over both W charges and including only the decay into electrons we compare our +predictions in Fig. 11. We show results at fixed order, at the resummed level, and at the +matched level. The effect of matching is large and we thus conclude that this value for the jet +veto is outside the sensible range for a purely resummed result, unlike for the Z study in the +previous subsection. +We observe excellent agreement with the theoretical prediction, albeit with a larger experimental +uncertainty. +The experimentally measured cross-section is 4.72 ± 0.30 nb while our best +prediction is 4.71+0.07 +−0.10 nb. Since this measurement corresponds to an integrated luminosity of +only 20 fb−1 it is clear that the high-luminosity LHC will eventually be able to provide a much +keener test of perturbative QCD in this process. +5.2 +W +W − production +Experimental studies of WW production were performed by both ATLAS [52, 53] and CMS [39, +54]. Here we focus on the CMS analysis of ref. [39] since it provides a measurement of the 0-jet +cross-section as a function of the jet pT veto. This cross-section measurment corresponds to +a sum over both electron and muon decays of the W bosons, which we denote by the label +pp → W −W + → 2ℓ2ν. In order to account for this in our calculation, we compute the result +for pp → e−µ+¯νeνµ at NNLO and multiply it by the factor that accounts exactly for all lepton +combinations through NLO. The impact of ZZ contributions in the same-flavor case results in +a slight enhancement over the naïve factor of four. We find that, independent of the value of +the jet veto in the range that we consider, this factor is equal to 4.15. +The CMS analysis only imposes a jet rapidity cut of ycut = 4.5, so our expectation is that the +standard jet veto resummation formalism should be appropriate for pveto +T +values between 60 +and 10 GeV, since in this case the logarithm of the ratio of Q to pveto +T +are in the range of 1.3 +to 3.1. This expectation is supported by the NNLO analysis in Table 6, which shows only a +– 21 – + +small 2% effect from the rapidity cut for pveto +T += 10 GeV (and none for values above that). +Unlike the processes considered so far, Q is no longer set by a resonance mass but is instead a +distribution with a peak slightly above the 2MW threshold. For illustration, we have used an +average value of Q ∼ 220 GeV. +We first fix the value of pveto +T += 30 GeV and study the sensitivity of the pure fixed-order and +resummed calculations to the jet-clustering parameter R. The results are shown in Fig. 12a. At +NLO, there is at most one additional parton, so the NLO result does not depend on the value of +R. However, the NNLL result exhibits a mild dependence on R, which is most noticeable in the +size of the uncertainties. These uncertainties are much larger for smaller values of R, as was +previously observed and discussed in the context of Higgs production in Ref. [12]. At NNLO, +the fixed-order calculation becomes sensitive to the value of R, although the dependence is very +small. At N3LLp, the dependence is reduced compared to NNLL, especially at small R. Overall, +these results suggest that the jet-clustering parameter has a mild effect on the predictions of +the fixed-order and resummed calculations for WW production. We have not investigated the +effect of small R resummation [14] on these results. +In Fig. 12b, we extend our previous analysis of the jet-veto dependence of WW production, +which was presented in Ref. [55]. The effect of matching is substantial for values of pveto +T +greater +than 20 GeV, so for typical jet vetoes in the range of 20 to 40 GeV, matched predictions are +important. We find that the fixed-order description is only capable of providing an adequate +result for the highest value of pveto +T +studied here. A comparison with the CMS measurement +shows better agreement with the matched resummed calculation, although the experimental +uncertainties are still substantial, corresponding to an integrated luminosity of 36 fb−1. +We eagerly anticipate a measurement with more statistics in order to hone this comparison. +Future measurements with higher precision and larger data samples will provide a more +stringent test of the theoretical predictions and help to refine our understanding of WW +production at the LHC. +5.3 +W ±Z production +5.3.1 +ATLAS +For W ±Z production, we first compare our results with an analysis from the ATLAS collabora- +tion at √s = 13 TeV [44]. The 0-jet cross-section is measured with jets defined by the anti-kT +Table 6: The pp → W −W + → 2ℓ2ν+0-jet cross-section at NNLO, with and without a jet +rapidity cut. +pveto +T +[GeV] +10 +25 +30 +35 +45 +60 +σ0−jet(no ycut) [fb] +535 +963 +1004 +1054 +1145 +1237 +σ0−jet(ycut = 4.5) [fb] +548 +963 +1004 +1054 +1145 +1237 +ϵ +0.02 +0.00 +0.00 +0.00 +0.00 +0.00 +– 22 – + +(a) Jet radius R dependence of fixed-order and +purely resummed results. +(b) Predictions and CMS measurement as a ratio +to the matched result. +Figure 12: Comparison of NNLO, N3LLp and matched N3LLp+NNLO results for W +W − +production. +qq' → W±Z, s = 13 TeV, ATLAS cuts, arXiv:1902.05759 + 31 ± 2.5 fb +29.7−1.2 ++0.9 fb +27 +30 +33 +36 +ATLAS +NLO +NNLL +NNLL+NLO +NNLO +N3LLp +N3LL+NNLO +σveto [pb] +Figure 13: Comparison of W ±Z jet-vetoed predictions with the ATLAS 13 TeV measurement +[44]. Shown are results at fixed order, purely resummed and matched. +algorithm with pT > 25 GeV, |y| < 4.5, and R = 0.4. +Since ln(Q/pveto +T +) = 2.3 (for pveto +T += 25 GeV, using an average Q of about 240 GeV), we expect +that standard jet veto resummation should be applicable in this case, since ycut = 4.5. We +have checked that the effect of the rapidity cut is at the per mille level, which is less than our +numerical precision. +The ATLAS result is presented for a single leptonic channel and summed over both W charges. +The corresponding theoretical predictions at fixed order, at the resummed level, and at the +matched level are shown in Fig. 13. +– 23 – + +qq' → W±Z, s = 13 TeV, CMS cuts, arXiv:2110.11231 + 166 ± 6 fb +128 ± 8 fb, ycut < ∞ +120 +140 +160 +180 +CMS +NLO +NNLL +NNLL+NLO +NNLO +N3LLp +N3LL+NNLO +σveto [pb] +Figure 14: Comparison of W ±Z jet-vetoed predictions with the CMS [45] 13 TeV measurement. +Shown are results at fixed-order, purely resummed and matched, all without a rapidity cut. +Overall, the measurement is in good agreement with both the N3LLp+NNLO and NNLO +predictions, within the mutual uncertainties. Only a more precise measurement would be +able to definitively support the need for resummation in this case. Since the ATLAS analysis +includes only 36 fb−1 of data, it is likely that a more precise measurement will be possible in +the near future. +5.3.2 +CMS +We now contrast the ATLAS study of the W ±Z process with one from CMS [45]. In the +CMS study, jets are defined by the anti-kT algorithm with pT > 25 GeV, |y| < 2.5, and +R = 0.4. +To assess the applicability of the jet-rapidity inclusive resummation framework, we must com- +pare ln(Q/pveto +T +) = 2.3 with ycut = 2.5. This suggests that the standard jet veto resummation +formalism may not be appropriate in this case, and that the use of ycut-dependent beam +functions [42] may be necessary to provide a reliable theoretical prediction. Despite this, we +still pursue the comparison here, without using ycut-dependent beam functions, to examine +the limitations of our approach. +The CMS result for W ±Z production is presented after summing over all lepton flavors and +both W charges. On the theoretical side, we perform a similar analysis, but ignore same-flavor +effects that only enter at the 2% level. To construct the jet-vetoed cross-section for the CMS +measurement, we combine the differential results in Figure 14(c) of Ref. [45] with the inclusive +cross-sections reported in Table 6 of the same reference. Our results are shown in Fig. 14. +We find that neither the resummed prediction nor the NNLO one are in good agreement +with the CMS data, even when the NNLO calculation takes the jet rapidity cut into account +(increasing the NNLO result from 128 fb to 137 fb). This suggests that resummation is required +in this case, and that the use of ycut-dependent beam functions is necessary to provide a reliable +theoretical prediction. Overall, these results highlight the importance of using appropriate +resummation techniques to accurately predict W ±Z production at the LHC with a small jet +rapidity cut. +– 24 – + +lepton cuts +ql1 +T > 20 GeV, ql2 +T > 10 GeV, +ql3,4 +T +> 5 GeV, |ηl| < 2.5 +lepton pair mass +60 GeV < ml−l+ < 120 GeV +jet veto +anti-kT , R = 0.5 +Table 7: Fiducial cuts used for the ZZ analysis, taken from the CMS study in Ref. [40]. +Table 8: The ZZ + 0-jet cross-section at NNLO (µ = Q), with and without a jet rapidity cut. +pveto +T +[GeV] +10 +20 +30 +40 +50 +60 +σ0−jet(no ycut) [fb] +13.3 +21.5 +25.8 +28.4 +30.3 +31.6 +σ0−jet(ycut = 4.5) [fb] +13.4 +21.5 +25.8 +28.4 +30.3 +31.6 +σ0−jet(ycut = 2.5) [fb] +14.9 +22.4 +26.3 +28.8 +30.6 +31.8 +ϵ(ycut = 4.5) +0.01 +0.00 +0.00 +0.00 +0.00 +0.00 +ϵ(ycut = 2.5) +0.12 +0.04 +0.02 +0.01 +0.01 +0.01 +5.4 +ZZ production +In the absence of jet-vetoed cross-sections for comparison, we use the cuts from a recent CMS +study [40] to investigate our theoretical predictions for ZZ production as a function of pveto +T +. +In the results that follow we consider a sum over Z decays into both electrons and muons, +which we denote by pp → ZZ → 4 leptons, and apply the cuts shown in Table 7. +We expect that standard jet veto resummation should provide good predictions for ycut = 4.5, +since ln(Q/pveto +T +) is in the range of 1.4 to 3.2 for pveto +T +values between 60 and 10 GeV, using an +average Q of about 240 GeV. For ycut = 2.5, we expect larger rapidity effects for the smallest +values of pveto +T +. This is supported by our analysis in Table 8, which shows only a very small +(1%) effect from a rapidity cut of ycut = 4.5 for pveto +T += 10 GeV (and no effect for higher values). +Even for ycut = 2.5, the rapidity cut has a relevant effect only for pveto +T +values below 30 GeV, +and is mostly insignificant beyond that. +Fig. 15a shows a comparison of the dependence on pveto +T +for purely-resummed results at two +different logarithmic orders. The central predictions are very similar at NNLL and N3LLp +and are consistent within uncertainties for all values of pveto +T +. Fig. 15b compares the matched +N3LLp+NNLO and NNLO results. The NNLO prediction has large uncertainties over the whole +range of pveto +T +and only overlaps with N3LLp+NNLO around 40 GeV and higher. The difference +between the central resummed and fixed-order results is significant (around 10%) for typical +values of pveto +T +around 30 GeV. For most relevant values of pveto +T +at the LHC, resummation is +clearly important for providing a precision prediction for this process. +5.5 +Higgs production +For gluon fusion Higgs production an important topic is the inclusion of finite top-quark mass +effects. Although at NNLO these could be included exactly [56, 57], the mass effects are not +relevant in the jet-vetoed case [58] at the current level of precision. A simple overall one-loop +– 25 – + +(a) Purely resummed results. +(b) Ratio to matched result. +Figure 15: Comparison of NNLO, N3LLp and matched N3LLp+NNLO results for ZZ production +as a function of the jet veto. +rescaling factor that takes into account the full mass dependence is sufficient to introduce mass +effects into mt → ∞ EFT predictions. In the resummation formalism, the coefficient for the +matching of Higgs production in QCD onto SCET can be calculated in two ways, referred to as +one-step and two-step procedures. +5.5.1 +One-step and two-step schemes +The one-step procedure is based on the observation that the ratio mH/mt is not large in a +logarithmic sense (c.f. ρ = m2 +H/m2 +t ≈ 1/2 and αs log 1/ρ ≈ 0.07). This procedure matches the +full QCD result, typically obtained at higher orders as an expansion in the parameter r, onto +SCET at the scale µh ∼ mH. In this way, terms of order ρ are retained, but logs of mt/mH +are neglected. +In the two-step procedure outlined in Refs. [59–62], the top quark is first integrated out at a +scale µt ≊ mt, and then the QCD effective Lagrangian is matched onto the SCET at a scale +µh ≊ mH. Running between µt and µh allows one to sum logarithms of mt/mH, and finite +top-mass effects are included by scaling the result by a correction factor obtained at leading +order (an increase with respect to the EFT result by a factor of 1.0653, see Eq. (G.5)). Terms +enhanced by powers of mH/mt are thus only included in an approximate fashion at NLO and +beyond. The one-step procedure is described in detail in Appendix G.1 and the two-step +procedure is described in Appendix G.2. +We compare the numerical difference between the one- and two-step schemes, computed at +√s = 13.6 TeV and for R = 0.4 in Fig. 16a. Guided by fixed-order results, and in accord with +– 26 – + +(a) Results in the one- or two-step scheme. The +lower panel shows the ratio of the one-step to the +two-step result. +(b) Results using a central scale of either µ2 +h = Q2 +or µ2 +h = −Q2. The lower panel shows the ratio of +the result for µ2 +h = −Q2 to the one for µ2 +h = Q2. +Figure 16: Comparison of NNLL and N3LLp predictions for Higgs production at √s = 13.6 TeV +as a function of the jet veto. +previous studies of this process [14], we set the hard (renormalization) scale using µh = Q/2. +We observe that the one-step scheme results in a cross-section that is about 1.7–2.3% larger at +NNLL and only 1.6% larger at N3LLp. This small difference occurs if one works rigorously at +a fixed order of αs. Working at a fixed order in αs in the component parts of the two-step +scheme can lead to larger differences, as described in more detail in Appendix G.3. +5.5.2 +Time-like vs. space-like µ2 +h +We now study the impact of choosing a time-like hard scale for the calculation of the Higgs +cross-section. To do this, we compare µ2 +h = (Q/2)2 (the space-like scale) with µ2 +h = −(Q/2)2 +(the time-like scale). The use of a time-like hard scale allows us to resum certain π2 terms, by +employing a complex strong coupling [51]. For this comparison, we consider purely resummed +results at NNLL and N3LLp accuracy. +Results are shown in Fig. 16b, for the two-step scheme computed at √s = 13.6 TeV with +R = 0.4. We observe that at NNLL, the resummation of the π2 terms significantly enhances +the cross-section by 17%. However, at N3LLp accuracy, this resummation only leads to a small +increase of 2% in the cross-section. +Results for the matched vetoed cross-section are shown in Fig. 17. +After matching, we +observe substantial agreement between the NNLO and N3LLp+NNLO calculations within +– 27 – + +Figure 17: Comparison of NNLL, N3LLp and N3LLp+NNLO predictions for Higgs production +at √s = 13.6 TeV as a function of the jet veto. +uncertainties. The central predictions differ by about 5% across the range, but the uncertainties +are substantially smaller in the resummed calculation. +6 +Conclusions +We have presented a comprehensive study of jet-veto resummation in the production of color +singlet final states using the most up-to-date theoretical ingredients and achieving N3LLp +accuracy. Our implementation in MCFM improves upon previous public NNLL calculations +by reducing theoretical uncertainties, as demonstrated by comparisons with ATLAS and CMS +results. Once the one remaining theoretical element, dveto +3 +, becomes available, it will be simple +to upgrade our predictions to full N3LL accuracy.2 +The primary motivation for this work comes from the need for reliable and accurate predictions +of jet-veto cross-sections in processes such as Higgs boson and W +W − production, which are +commonly used to study new physics at the LHC. In these processes, the imposition of a jet +veto is often necessary to suppress backgrounds and enhance sensitivity to new physics signals. +Experimental results going beyond these two processes are much less frequent. We encourage +the experimental collaborations to consider measurements of more Standard Model processes +with a jet veto, as larger data samples become available, to better understand the dependence +of these processes on the jet veto parameters pveto +T +and R. +2We have shown that the effect of including gluon-induced process in W +W − and ZZ production is +numerically a small effect, so that NLL accuracy is sufficient for these sub-processes. +– 28 – + +In addition to providing improved predictions for jet-veto cross-sections, our work also serves +as a valuable tool for testing and validation of general purpose shower Monte Carlo programs. +Our code allows for a detailed investigation of the dependence on the jet parameters pveto +T +and +R, providing a benchmark for assessing the logarithmic accuracy and reliability of Monte Carlo +simulations in this important class of processes. +Our analysis shows that at the currently experimentally used values of pveto +T +in W and Z +production, the logarithms are not large enough to justify the use of jet-veto resummation. +In these cases, fixed-order perturbation theory, which can be used to give the results with +a jet veto over a limited range of rapidities, is simpler and sufficient. We have also found +that attempts to resum π2 terms using a timelike renormalization point have little numerical +importance at N3LLp if the pveto +T +scale is around 20 to 30 GeV. +The production of a Higgs boson is an exception among single-boson processes. In this case, +the combination of larger corrections from color factors and slightly larger values of the scale +(mH) appearing in the jet veto logarithms make resummation an important tool for improving +the accuracy of predictions. In the appendix we have investigated the differences between the +one-step and two-step procedures for calculating the hard function at the scale of pveto +T +. We +find agreement within 2% of these two approaches. +The W +W − production process, where the jet veto has experimental importance, requires +both resummation and matching to NNLO. For the ZZ process resummation is mandatory +but the matching to fixed order is less important. Although this reflects the expectation that +the resummed prediction is more accurate for systems of higher invariant mass, these findings +depend on the exact nature of the cuts for each process. Our work provides a comprehensive +theoretical framework for studying jet vetoes in vector boson pair processes, and as data +becomes available, a comparative experimental study would be of great interest and could help +to validate our theoretical predictions. +Acknowledgments +RKE would like to thank Simone Alioli, Thomas Becher, Andrew Gilbert, Pier Monni and +Philip Sommer for useful discussions. In addition, RKE would like to thank TTP in Karlsruhe +for hospitality during the drafting of this paper. TN would like to thank Robert Szafron +for useful discussions. SS is supported in part by the SERB-MATRICS under Grant No. +MTR/2022/000135. This manuscript has been authored by Fermi Research Alliance, LLC +under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of +Science, Office of High Energy Physics. This research used resources of the Wilson High- +Performance Computing Facility at Fermilab. This research also used resources of the National +Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office +of Science User Facility located at Lawrence Berkeley National Laboratory, operated under +Contract No. DE-AC02-05CH11231 using NERSC award HEP-ERCAP0021890. +– 29 – + +A +Reduced beam functions +We have used the two loop beam function in the presence of a jet veto calculated in Ref. [24]. +Their calculation, together with the corresponding soft function [25] has been performed in +SCET using the exponential rapidity regulator [27]. The beam function for quark initiated +processes in the presence of a jet veto has also been presented in Mellin space in Ref. [63]. +The calculation in Ref. [24] has a perturbative expansion, +Iij = +∞ +� +k=0 +�αs +4π +�k +I(k) +ij . +(A.1) +The beam functions with a jet veto are decomposed into a reference observable, the beam +function for the transverse momentum of a color singlet observable and a remainder term +accounting for the effects of jet clustering, +Iij(x, Q, pveto +T +, R; µ, ν) = I⊥ +ij(x, Q, pveto +T +; µ, ν) + ∆Iij(x, Q, pveto +T +, R; µ, ν) . +(A.2) +Since the divergence structure of the reference observable is the same as the beam function +with a jet veto, ∆Iij can be calculated in four dimensions. Results for the reference observable +are available in Refs. [64, 65]. +The reduced beam function kernels ¯I as used in our setup are extracted from the coefficient I +as +¯Iij(z, pveto +T +, R, µ) = e−hA(pveto +T +,µ) Iij(z, pveto +T +, R, µ) . +(A.3) +They similarly follow a perturbative expansion +¯Iik(z, pveto +T +, R, µ) = δik δ(1−z)+ αs +4π +¯I(1) +ik (z, pveto +T +, µ)+ +�αs +4π +�2 ¯I(2) +ik (z, pveto +T +, R, µ)+O(α3 +s) . (A.4) +Contributions at order αs +The αs contributions to ¯I were first obtained in Refs. [3, 50] and read, +¯Iij(z, pveto +T +, R, µ) = δ(1 − z) δij + αs +4π +� +−2P (1) +ij (z) L⊥ + R(1) +ij (z) +� ++ O(α2 +s) , +(A.5) +where L⊥ = 2 ln(µ/pveto +T +). R is the jet measure used in Eq. (2.1) and R(1)(z) is a remainder +function given below. At this order there is no dependence on the jet radius, R. +Throughout this paper we expand in powers of αs/(4π). The one exception to this rule are +the perturbative DGLAP splitting functions, +Pij(z) = αs +2πP (1)(z) + +�αs +2π +�2 +P (2)(z) + . . . +(A.6) +– 30 – + +Explicit expressions for P (1) and P (2) are given in Appendices C.2 and C.3. The remainder +functions at order αS are [66] +R(1) +qq (z) = CF +� +2(1 − z) − π2 +6 δ(1 − z) +� +, +R(1) +qg (z) = 4TF z(1 − z) , +R(1) +gg (z) = −CA +π2 +6 δ(1 − z) , +R(1) +gq (z) = 2CF z . +(A.7) +where CA = 3, CF = 4 +3, TF = 1 +2. +Contributions at order α2 +s +At order α2 +s we have +¯I(2) +ik (z, pveto +T +, R, µ) = +� +2P (1) +ij (x) ⊗ P (1) +jk (y) − β0P (1) +ik (z) +� +L2 +⊥ ++ +� +− 4P (2) +ik (z) + β0R(1) +ik (z) − 2R(1) +ij (x) ⊗ P (1) +jk (y) +� +L⊥ + R(2) +ik (z, R) . +(A.8) +In this equation ⊗ represents a convolution, +f(x) ⊗ g(y) = +� 1 +0 +dx +� 1 +0 +dyf(x) g(y) δ(z − xy) = +� 1 +z +dy +y f(z/y) g(y) . +(A.9) +Explicit expressions for P (1) and P (2) are given in Appendices C.2 and C.3. The expressions +for P (1) ⊗ P (1), R(1) ⊗ P (1) are given in appendix C.4. +The results from Refs. [24, 25] recast in the language of reduced beam functions allow us to +extract R(2) +ik (z, R). We have checked that the reduced beam functions have the form predicted +by Eqs. (A.5) and (A.8). In addition, we have confirmed the known results for the α2 +s R- +dependent contribution to the collinear anomaly exponent. The result for the collinear anomaly +exponent is given in Section 2.1. +A.1 +Structure of the two-loop reduced beam function +While a numerical evaluation of the analytical formulas for the reduced beam functions is +possible, we choose to perform a spline interpolation for improved numerical efficiency. +The reduced beam functions contain distributions of the following structure, +¯I(2) +ij (z, pveto +T +, R, µ) = ¯I(2) +ij,−1(pveto +T +, R, µ) δ(1 − z) + ¯I(2) +ij,0(pveto +T +, R, µ) D0(1 − z) ++ ¯I(2) +ij,1(pveto +T +, R, µ) D1(1 − z) + ¯I(2) +ij,2(z, pveto +T +, R, µ) , +(A.10) +where, +D0(1 − z) = +1 +[1 − z]+ +, +D1(1 − z) = +�ln(1 − z) +(1 − z) +� ++ +. +(A.11) +¯I(2) +ij,2(z, pveto +T +, R, µ) contains terms which are regular at z = 1. +– 31 – + +The analytic results for the beam function of Ref. [24] are presented as a power series in +R up to powers of R8. The functions themselves contain powers of 1/(1 − z)n, in certain +cases up to n = 7 or 8. However, these singularities at z = 1 are fictitious as can be seen by +explicit expansion. The beam functions require special treatment in this region for numerical +stability. +The dominant region in the convolution of the function ¯I with the parton distributions is +precisely the region z ∼ 1. If we assume a parton distribution f(x) ∼ 1/x we have, +¯I ⊗ f = +� 1 +x +dz +z +¯I(z) f(x/z) ∼ 1 +x +� 1 +x +dz ¯I(z) , +(A.12) +showing that all regions of z contribute equally to the integral. However if, as expected, the +parton distribution function falls off more rapidly as x → 1, say f(x) ∼ (1 − x)n/x, +¯I ⊗ f = +� 1 +x +dz +z +¯I(z) f(x/z) ∼ 1 +x +� 1 +x +dz ¯I(z) (1 − x/z)n . +(A.13) +Thus, it is precisely the large values of z which are crucial for the integral. In other words, +the parton shower process is dominated by cascade from nearby values of x. Larger cascades +from more distant points are suppressed by the fall-off of the parton distributions. In view of +the importance of the region z = 1, for numerical stability we perform an expansion about +z = 1. +The absolute value of R(2) for the various parton transitions is shown in Fig. 18. Individual +R-dependent terms contain expressions of the form R2n/(1 − z)k where k can be a high power. +However, the singularity at z = 1 is only apparent. The resultant limiting forms obtained by +series expansion about z = 1 are shown by the dashed lines in the figures. In practice, we +switch to the expanded form at z = 0.9, although the figures demonstrate that the expanded +forms are accurate down to much smaller values of z. +B +Definition of the beta function and anomalous dimensions +The coefficients βn, ΓA +n and γg +n have perturbative expansions in powers of the renormalized +coupling. Details are presented below. +B.1 +Expansion of β-function +The beta function is defined as, +dαs(µ) +d ln µ += β(µ) = −2αs(µ) +∞ +� +n=0 +βn +�αs +4π +�n+1 += −2αs(µ) αs(µ) +4π +� +β0 + β1 +αs(µ) +4π ++ β2 +�αs(µ) +4π +�2 ++ β3 +�αs(µ) +4π +�3 ++ . . . +� +. +(B.1) +– 32 – + +(a) gg case. +(b) qq case +(c) gq case. +(d) qg case +(e) ¯qq case. +(f) q′q case. +Figure 18: Absolute value of R(2) for jet measure R = 0.5. The ¯q′q case is the same as the +q′q case. The sign of the contribution in the various regions is indicated. +The coefficients of the MS β function to four loops are [67–69], +β0 = 11 +3 CA − 4 +3 TF nf , +– 33 – + +β1 = 34 +3 C2 +A − +�20 +3 CA + 4CF +� +TF nf , +β2 = 2857 +54 C3 +A + +� +C2 +F − 205 +18 CF CA − 1415 +54 C2 +A +� +2TF nf + +�11 +9 CF + 79 +54 CA +� +4T 2 +F n2 +f , +β3 = C4 +A +�150653 +486 +− 44 +9 ζ3 +� ++ C3 +ATF nf +� +−39143 +81 ++ 136 +3 ζ3 +� ++C2 +ACF TF nf +�7073 +243 − 656 +9 ζ3 +� ++ CAC2 +F TF nf +� +−4204 +27 ++ 352 +9 ζ3 +� ++46C3 +F TF nf + C2 +AT 2 +F n2 +f +�7930 +81 ++ 224 +9 ζ3 +� ++ C2 +F T 2 +F n2 +f +�1352 +27 +− 704 +9 ζ3 +� ++CACF T 2 +F n2 +f +�17152 +243 ++ 448 +9 ζ3 +� ++ 424 +243CAT 3 +F n3 +f + 1232 +243 CF T 3 +F n3 +f ++dabcd +A +dabcd +A +NA +� +−80 +9 + 704 +3 ζ3 +� ++ nf +dabcd +F +dabcd +A +NA +�512 +9 +− 1664 +3 +ζ3 +� ++n2 +f +dabcd +F +dabcd +F +NA +� +−704 +9 ++ 512 +3 ζ3 +� +. +(B.2) +For the normalization of the SU(N) generators, the conventions of Refs. [69, 70] are, +dabcd +A +dabcd +A +NA += N2(N2 + 36) +24 +, +dabcd +F +dabcd +A +NA += N(N2 + 6) +48 +, +dabcd +F +dabcd +F +NA += N4 − 6N2 + 18 +96N2 +, +NA = N2 − 1 , +NF = N. +(B.3) +Numerical values for the β-function coefficients are, +β0 = 11 − 2 +3 nf , +β1 = 102 − 38 +3 nf , +β2 = 2857 +2 +− 5033 +18 nf + 325 +54 n2 +f , +β3 = 149753 +6 ++ 3564ζ3 − +�1078361 +162 ++ 6508 +27 ζ3 +� +nf + +�50065 +162 ++ 6472 +81 ζ3 +� +n2 +f ++ 1093 +729 n3 +f . +(B.4) +B.2 +Cusp Anomalous Dimension +The cusp anomalous dimension depends on the label B which takes the two values, B = A, F +for gluons and quarks, respectively. Its perturbative expansion is, +ΓB +cusp(µ) = +∞ +� +n=0 +ΓB +n +�αs +4π +�n+1 +. +(B.5) +– 34 – + +The coefficients up to four loops are [71, 72], +ΓB +0 = 4CB , +(B.6) +ΓB +1 = 16CB +� +(CA +�67 +36 − π2 +12 +� +− 5 +9nfTF +� +, +(B.7) +ΓB +2 = 64CB +� +C2 +A +�11ζ3 +24 + 245 +96 − 67π2 +216 + 11π4 +720 +� ++ nfTF CF +� +ζ3 − 55 +48 +� ++ nfTF CA +� +−7ζ3 +6 − 209 +216 + 5π2 +54 +� +− 1 +27(nfTF )2 +� +, +(B.8) +ΓB +3 = 256CB +� +C3 +A +�1309ζ3 +432 +− 11π2ζ3 +144 +− ζ2 +3 +16 − 451ζ5 +288 + 42139 +10368 − 5525π2 +7776 ++ 451π4 +5760 − 313π6 +90720 +� ++ nfTF C2 +A +� +−361ζ3 +54 ++ 7π2ζ3 +36 ++ 131ζ5 +72 +− 24137 +10368 + 635π2 +1944 − 11π4 +2160 +� ++ nfTF CF CA +�29ζ3 +9 +− π2ζ3 +6 ++ 5ζ5 +4 − 17033 +5184 + 55π2 +288 − 11π4 +720 +� ++ nfTF C2 +F +�37ζ3 +24 − 5ζ5 +2 + 143 +288 +� ++ (nfTF )2CA +�35ζ3 +27 − 7π4 +1080 − 19π2 +972 + 923 +5184 +� ++ (nfTF )2CF +� +−10ζ3 +9 ++ π4 +180 + 299 +648 +� ++ (nfTF )3 +� +− 1 +81 + 2ζ3 +27 +� � ++ 256dabcd +B +dabcd +A +NB +�ζ3 +6 − 3ζ2 +3 +2 ++ 55ζ5 +12 − π2 +12 − 31π6 +7560 +� ++ 256nf +dabcd +B +dabcd +F +NB +�π2 +6 − ζ3 +3 − 5ζ5 +3 +� +. +(B.9) +In addition to the relations in Eq. (B.3) we need the related quantities, +dabcd +F +dabcd +A +NF += (N2 − 1)(N2 + 6) +48 +, +dabcd +F +dabcd +F +NF += (N2 − 1)(N4 − 6N2 + 18) +96N3 +. +(B.10) +B.3 +Non-cusp anomalous dimension +The non-cusp anomalous dimension has the expansion, +γq,g(µ) = +∞ +� +n=0 +γq,g +n +�αs +4π +�n+1 +. +(B.11) +– 35 – + +We take the coefficients up to three loops from ref. [73] Eq. I.4, +γq +0 = −3CF , +(B.12) +γq +1 = C2 +F +� +2π2 − 3 +2 − 24ζ3 +� ++ CF CA +� +26ζ3 − 961 +54 − 11π2 +6 +� ++ CF TF nf +�130 +27 + 2π2 +3 +� +, +(B.13) +γq +2 = C3 +F +� +− 29 +2 − 3π2 − 8π4 +5 +− 68ζ3 + 16π2 +3 +ζ3 + 240ζ5 +� ++ C2 +F CA +� +− 151 +4 ++ 205π2 +9 ++ 247π4 +135 +− 844 +3 ζ3 − 8π2 +3 ζ3 − 120ζ5 +� ++ CF C2 +A +� +− 139345 +2916 +− 7163π2 +486 +− 83π4 +90 ++ 3526 +9 +ζ3 − 44π2 +9 +ζ3 − 136ζ5 +� ++ C2 +F TF nf +�2953 +27 +− 26π2 +9 +− 28π4 +27 ++ 512 +9 ζ3 +� ++ CF CATF nf +� +− 17318 +729 ++ 2594π2 +243 ++ 22π4 +45 +− 1928 +27 ζ3 +� ++ CF T 2 +F n2 +f +�9668 +729 − 40π2 +27 +− 32 +27ζ3 +� +. +(B.14) +From ref. [74], Eq A5 we take, +γg +0 = −β0 , +(B.15) +γg +1 = C2 +A +�11π2 +18 +− 692 +27 + 2ζ3 +� ++ CATF nf +�256 +27 − 2π2 +9 +� ++ 4CF TF nf += C2 +A +� +2ζ3 − 59 +9 +� ++ CAβ0 +�π2 +6 − 19 +9 +� +− β1 , +(B.16) +γg +2 = C3 +A +� +− 97186 +729 ++ 6109π2 +486 +− 319π4 +270 ++ 122 +3 ζ3 − 20π2 +9 +ζ3 − 16ζ5 +� ++ C2 +ATF nf +�30715 +729 +− 1198π2 +243 ++ 82π4 +135 + 712 +27 ζ3 +� ++ CACF TF nf +�2434 +27 +− 2π2 +3 +− 8π4 +45 − 304 +9 ζ3 +� +− 2C2 +F TF nf + CAT 2 +F n2 +f +� +− 538 +729 + 40π2 +81 +− 224 +27 ζ3 +� +− 44 +9 CF T 2 +F n2 +f . +(B.17) +Primary references for the calculation of these coefficients can be found in Ref. [73]. +We now present results for γS and γt which are needed for the implementation of the two-step +calculation of the hard function for Higgs boson production. Following Ref. [61] we have, for +the first three expansion coefficients of the anomalous dimension γS that enters the evolution +– 36 – + +equation of the hard matching coefficient CS (see also [59, 60]), +γS +0 = 0 , +(B.18) +γS +1 = C2 +A +� +−160 +27 + 11π2 +9 ++ 4ζ3 +� ++ CATF nf +� +−208 +27 − 4π2 +9 +� +− 8CF TF nf , +(B.19) +γS +2 = C3 +A +�37045 +729 ++ 6109π2 +243 +− 319π4 +135 ++ +�244 +3 +− 40π2 +9 +� +ζ3 − 32ζ5 +� ++ C2 +ATF nf +� +−167800 +729 +− 2396π2 +243 ++ 164π4 +135 ++ 1424 +27 ζ3 +� ++ CACF TF nf +�1178 +27 +− 4π2 +3 +− 16π4 +45 +− 608 +9 ζ3 +� ++ 8C2 +F TF nf ++ CAT 2 +F n2 +f +�24520 +729 ++ 80π2 +81 +− 448 +27 ζ3 +� ++ 176 +9 CF T 2 +F n2 +f . +(B.20) +The function γt is given by, +γt(αs) = α2 +s +d +dαs +� +β(αs) +α2s +� += −2β1 +�αs +4π +�2 +− 4β2 +�αs +4π +�3 +− 6β3 +�αs +4π +�4 ++ O(α5 +s) . +(B.21) +As shown in Eq. (G.22) µ independence provides the constraint, +2γg(αs) = γt(αs) + γS(αs) + β(αs)/αs , +(B.22) +leading to the simple relationship between the coefficients in γg and γS, +γS +0 = 2γg +0 + 2β0 , γS +1 = 2γg +1 + 4β1 , γS +2 = 2γg +2 + 6β2, γS +3 = 2γg +3 + 8β3 . +(B.23) +C +Definitions for beam function ingredients +C.1 +Exponent h +We define the auxiliary functions hB for B = F, A which, when combined with the hard +function and the collinear anomaly factor, will yield a renormalization group invariant hard +function. hF/A is defined to satisfy the RGE equation, +d +d ln µ hF/A(pveto +T +, µ) = 2 ΓF/A +cusp(µ) ln +µ +pveto +T +− 2 γq/g(µ) , +(C.1) +The factor h removes logarithms from the beam function and has a perturbative expansion in +terms of the renormalized coupling, +hB(pveto +T +, µ) = αs +4πhB +0 + +�αs +4π +�2 +hB +1 + +�αs +4π +�3 +hB +2 + +�αs +4π +�4 +hB +3 + . . . . +(C.2) +– 37 – + +Thus for the particular case B = F we have that, +hF +0 (pveto +T +, µ) = 1 +4ΓF +0 L2 +⊥ − γq +0L⊥ , +hF +1 (pveto +T +, µ) = 1 +12ΓF +0 β0L3 +⊥ + 1 +4(ΓF +1 − 2γq +0β0)L2 +⊥ − γq +1L⊥ , +hF +2 (pveto +T +, µ) = 1 +24ΓF +0 β2 +0L4 +⊥ + ( 1 +12ΓF +0 β1 + 1 +6ΓF +1 β0 − 1 +3γq +0β2 +0)L3 +⊥ ++ (1 +4ΓF +2 − 1 +2γq +0β1 − γq +1β0)L2 +⊥ − γq +2L⊥ , +hF +3 (pveto +T +, µ) = + 1 +40ΓF +0 β3 +0L5 +⊥ + ( 5 +48ΓF +0 β0β1 + 1 +8ΓF +1 β2 +0 − 1 +4γq +0β3 +0)L4 +⊥ ++ ( 1 +12ΓF +0 β2 + 1 +6ΓF +1 β1 + 1 +4ΓF +2 β0 − 5 +6γq +0β0β1 − γq +1β2 +0)L3 +⊥ ++ (1 +4ΓF +3 − 1 +2γq +0β2 − γq +1β1 − 3 +2γq +2β0)L2 +⊥ − γq +3L⊥ , +(C.3) +where L⊥ = 2 ln(µ/pveto +T +). The corresponding result for B = A, q = g, (i.e. for incoming gluons) +is given by a similar expression mutatis mutandis. The expansion coefficients of the β-function, +ΓF/A +cusp and γq/g, used in Eq. (C.3), are as given in Appendices B.1,B.2 and B.3. +C.2 +One loop splitting functions +The one-loop DGLAP splitting functions as defined in [75] are +P (1) +qq (z) = CF +�1 + z2 +1 − z +� ++ +, +(C.4) +P (1) +qg (z) = TF +� +z2 + (1 − z)2� +, +(C.5) +P (1) +gg (z) = 2CA +� +z +(1 − z)+ ++ 1 − z +z ++ z(1 − z) +� ++ β0 +2 δ(1 − z) , +(C.6) +P (1) +gq (z) = CF +1 + (1 − z)2 +z +, +(C.7) +C.3 +Two loop splitting functions +Now we turn to the two-loop anomalous dimensions that contribute at sub-leading log level to +the transitions between parton types. In the quark sector there are four independent transitions +that we must produce values for (viz. q′ ← q,¯q′ ← q,q ← q and ¯q ← q). They are expressed in +terms of four functions, +P (2) +q′q = P S(2) +qq +, P (2) +¯q′q = P S(2) +¯qq +, P (2) +qq = P V (2) +qq ++ P S(2) +qq +, P (2) +¯qq = P V (2) +¯qq ++ P S(2) +¯qq +. +(C.8) +– 38 – + +At next-to-leading order, the functions P S +qq and P S +¯qq are non-zero, but we have the additional +relation, P S +qq = P S +¯qq. To facilitate the presentation we define the auxiliary functions, +pqq(z) = +2 +1 − z − 1 − z , p(r) +qq (z) = −1 − z , +(C.9) +pqg(z) = z2 + (1 − z)2 , +(C.10) +pgq(z) = 1 + (1 − z)2 +z +, +(C.11) +pgg(z) = +1 +1 − z + 1 +z − 2 + z(1 − z), p(r) +gg (z) = 1 +z − 2 + z(1 − z) . +(C.12) +The two valence functions needed for the quark sector are, [76–78], +P V (2) +qq +(z) = C2 +F +� +− +� +2 ln z ln(1 − z) + 3 +2 ln z +� +pqq(z) +− +� +3 +2 + 7 +2z +� +ln z − 1 +2(1 + z) ln2 z − 5(1 − z) +� ++CF CA +� +(1 + z) ln z + 20 +3 (1 − z) + +� +1 +2 ln2 z + 11 +6 ln z +� +pqq(z) ++ +� +67 +18 − π2 +6 +�� +1 +(1 − z)+ ++ p(r) +qq (z) +�� +−CF TF nf +� +4 +3(1 − z) + 2 +3pqq(z) ln z + 10 +9 +� +1 +(1 − z)+ ++ p(r) +qq (z) +�� ++ +� +C2 +F +� +3 +8 − π2 +2 + 6ζ3 +� ++ CF CA +� +17 +24 + 11π2 +18 +− 3ζ3 +� +−CF TF nf +� +1 +6 + 2π2 +9 +�� +δ(1 − z) , +(C.13) +P V (2) +¯qq +(z) = CF +� +CF − CA +2 +�� +2pqq(−z)S2(z) + 2(1 + z) ln z + 4(1 − z) +� +, +(C.14) +and for the singlet function we have, +P S(2) +qq += CF TF +� +20 +9z − 2 + 6z − 56 +9 z2 + (1 + 5z + 8 +3z2) ln z − (1 + z) ln2 z +� +. +(C.15) +The other three transitions are simply given by, +P (2) +qg = CF TF +� +2 − 9 +2z − (1 +2 − 2z) ln z − (1 +2 − z) ln2 z + 2 ln(1 − z) +– 39 – + ++ +� +ln2 +� +1 − z +z +� +− 2 ln +� +1 − z +z +� +− π2 +3 + 5 +� +pqg(z) +� ++CATF +� +91 +9 + 7 +9z + 20 +9z + +� +68 +3 z − 19 +3 +� +ln z +−2 ln(1 − z) − (1 + 4z) ln2 z + pqg(−z)S2(z) ++ +� +− 1 +2 ln2 z + 22 +3 ln z − ln2(1 − z) + 2 ln(1 − z) + π2 +6 − 109 +9 +� +pqg(z) +� +, +(C.16) +P (2) +gq (z) = C2 +F +� +− 5 +2 − 7z +2 + +� +2 + 7 +2z +� +ln z − +� +1 − 1 +2z +� +ln2 z +− 2z ln(1 − z) − +� +3 ln(1 − z) + ln2(1 − z) +� +pgq(z) +� ++CF CA +� +28 +9 + 65 +18z + 44 +9 z2 − +� +12 + 5z + 8 +3z2 +� +ln z ++(4 + z) ln2 z + 2z ln(1 − z) + S2(z)pgq(−z) ++ +� +1 +2 − 2 ln z ln(1 − z) + 1 +2 ln2 z + 11 +3 ln(1 − z) + ln2(1 − z) − π2 +6 +� +pgq(z) +� ++CF TF nf +� +− 4 +3z − +� +20 +9 + 4 +3 ln(1 − z) +� +pgq(z) +� +, +(C.17) +P (2) +gg (z) = CF TF nf +� +− 16 + 8z + 20 +3 z2 + 4 +3z − (6 + 10z) ln z − (2 + 2z) ln2 z +� ++CATF nf +� +2 − 2z + 26 +9 +� +z2 − 1 +z +� +− 4 +3(1 + z) ln z +−20 +9 +� +1 +(1 − z)+ ++ p(r) +gg (z) +�� ++C2 +A +� +27 +2 (1 − z) + 67 +9 +� +z2 − 1 +z +� +− +� +25 +3 − 11 +3 z + 44 +3 z2 +� +ln z ++4(1 + z) ln2 z + 2pgg(−z)S2(z) ++ +� +ln2 z − 4 ln z ln(1 − z) +� +pgg(z) + +� +67 +9 − π2 +3 +�� +1 +(1 − z)+ ++ p(r) +gg (z) +�� ++ +� +C2 +A +�8 +3 + 3ζ3 +� +− CF TF nf − 4 +3CATF nf +� +δ(1 − z) . +(C.18) +– 40 – + +The function S2(z) is defined by +S2(z) = +� +1 +1+z +z +1+z +dy +y ln +�1 − y +y +� +. +(C.19) +In terms of the dilogarithm function +Li2(z) = − +� z +0 +dy +y ln(1 − y) , +(C.20) +we have +S2(z) = −2 Li2(−z) + 1 +2 ln2 z − 2 ln z ln(1 + z) − π2 +6 . +(C.21) +C.4 +P (1) ⊗ P (1) and R(1) ⊗ P (1) +We give here expressions for the convolutions of functions appearing in the beam functions. +The convolutions are defined as in Eq. (A.9). Similar expressions have been given in [1, 12] +The convolutions of the one-loop DGLAP kernels from Eqs. (C.4) are, +P (1) +qq +⊗ P (1) +qg = CF TF +� +2z − 1 +2 + (2z − 4z2 − 1) ln z + (2 − 4z(1 − z)) ln(1 − z) +� +, +(C.22) +P (1) +qg +⊗ P (1) +gg = CATF +� +2(1 + 4z) ln z + 4 +3z + 1 + 8z − 31 +3 z2� ++ +� +2CA ln(1 − z) + β0 +2 +� +P (1) +qg (z) , +(C.23) +P (1) +gq ⊗ P (1) +qq = C2 +F +� +2 − 1 +2z + (2 − z) ln z +� ++ 2CF P (1) +gq (z) ln(1 − z) +� +, +(C.24) +P (1) +gg ⊗ P (1) +gq = CACF +� +8 + z + (4z3 − 31) +3z +− 4(1 + z + z2) +z +ln z +� ++ +� +2CA ln(1 − z) + β0 +2 +� +P (1) +gq (z) , +(C.25) +P (1) +qg +⊗ P (1) +gq = CF TF +� +2(1 + z) ln z + 1 − z + 4 +3 +(1 − z3) +z +� +, +(C.26) +P (1) +qq +⊗ P (1) +qq = C2 +F +� +8 +�ln(1 − z) +(1 − z) +� ++ − 4(1 + z) ln(1 − z) − 2(1 − z) ++ +� +3 + 3z − +4 +(1 − z) +� +ln z +� ++ 3CF P (1) +qq (z) − C2 +F (9 +4 + 4ζ2)δ(1 − z) , +(C.27) +P (1) +gg ⊗ P (1) +gg = 4C2 +A +� +2 +�ln(1 − z) +(1 − z) +� ++ + 2((1 − z) +z ++ z(1 − z) − 1) ln(1 − z) + 3(1 − z) +− ( +1 +1 − z + 1 +z − z2 + 3z) ln z − 11(1 − z3) +3z +� ++ β0P (1) +gg (z) − (β2 +0 +4 + 4C2 +Aζ2)δ(1 − z) . +(C.28) +The convolutions of lowest order DGLAP kernels, Eq. (C.4) with the one-loop finite terms in +the beam functions, Eq. (A.7) are, +R(1) +gg ⊗ P (1) +gg = −CAζ2P (1) +gg (z) , +(C.29) +– 41 – + +R(1) +gq ⊗ P (1) +qg += 2CF TF +� +(1 − z)(1 + 2z) + 2z ln z +� +, +(C.30) +R(1) +qq ⊗ P (1) +qq += CF +� +CF (1 − z)(4 ln(1 − z) − 2 ln z − 1) − ζ2P (1) +qq (z) +� +, +(C.31) +R(1) +qg ⊗ P (1) +gq = −4CF TF +� +1 + z ln z − (1 + 2z3) +3z +� +, +(C.32) +R(1) +qg ⊗ P (1) +gg = −CATF (16z ln z − 68 +3 z2 + 20z + 4 − 4 +3z ) ++ (2CA ln(1 − z) + β0 +2 )R(1) +qg (z) , +(C.33) +R(1) +qq ⊗ P (1) +qg += CF TF (2z2 + 2z − 4 − (2 + 4z) ln z) − CF ζ2P (1) +qg (z) , +(C.34) +R(1) +gq ⊗ P (1) +qq += −C2 +F (2z ln z − 4z ln(1 − z) − z − 2) , +(C.35) +R(1) +gg ⊗ P (1) +gq = −CAζ2P (1) +gq (z) . +(C.36) +D +Rapidity anomalous dimension +Solving the collinear anomaly RG equation (Eq. (2.13)) as an expansion in αs (Eq. (2.15)) we +have that, +F (0) +gg (pveto +T +, µh) = ΓA +0 L⊥ + dveto +1 +(R, A) , +F (1) +gg (pveto +T +, µh) = 1 +2ΓA +0 β0L2 +⊥ + ΓA +1 L⊥ + dveto +2 +(R, A) , +F (2) +gg (pveto +T +, µh) = 1 +3ΓA +0 β2 +0L3 +⊥ + 1 +2(ΓA +0 β1 + 2ΓA +1 β0)L2 +⊥ ++ (ΓA +2 + 2β0dveto +2 +(R, A))L⊥ + dveto +3 +(R, A) , +F (3) +gg (pveto +T +, µh) = 1 +4β3 +0ΓA +0 L4 +⊥ + (ΓA +1 β2 +0 + 5 +6ΓA +0 β0β1)L3 +⊥ ++ (1 +2ΓA +0 β2 + ΓA +1 β1 + 3 +2ΓA +2 β0 + 3dveto +2 +(R, A)β2 +0)L2 +⊥ ++ (ΓA +3 + 3dveto +3 +(R, A)β0 + 2dveto +2 +(R, A)β1)L⊥ + dveto +4 +(R, A) . +(D.1) +where L⊥ = 2 ln(µh/pveto +T +). The corresponding result for Fqq is given in Eq. (2.16). Because +Fgg appears in the exponent, we see that dveto +1 +contributes in NLL, dveto +2 +in NNLL, and dveto +3 +in +N3LL. +– 42 – + +D.1 +dveto +2 +expansion +The expansion coefficients for dveto +2 +, which is defined in Eq. (2.18), are given by [4, 5, 12], +cA +L = 131 +72 − π2 +6 − 11 +6 ln 2 = −1.096259 , +cA +0 = −805 +216 + 11π2 +72 ++ 35 +18 ln 2 + 11 +6 ln2 2 + ζ3 +2 = 0.6106495 , +cA +2 = +1429 +172800 + π2 +48 + 13 +180 ln 2 = 0.263947 , +cA +4 = − 9383279 +406425600 − +π2 +3456 + +587 +120960 ln 2 = −0.0225794 , +cA +6 = +74801417 +97542144000 − +23 +67200 ln 2 = 5.29625 · 10−4 , +cA +8 = − +50937246539 +2266099089408000 − +π2 +24883200 + +28529 +1916006400 ln 2 = −1.25537 · 10−5 , +cA +10 = +348989849431 +243708656615424000 − +3509 +3962649600 ln 2 = 8.18201 · 10−7 . +(D.2) +and +cf +L = −23 +36 + 2 +3 ln 2 = −0.1767908 , +cf +0 = 157 +108 − π2 +18 − 8 +9 ln 2 − 2 +3 ln2 2 = −0.03104049 , +cf +2 = 3071 +86400 − +7 +360 ln 2 = 0.0220661 , +cf +4 = − +168401 +101606400 + +53 +30240 ln 2 = −4.42544 · 10−4 , +cf +6 = +7001023 +48771072000 − +11 +100800 ln 2 = 6.79076 · 10−5 , +cf +8 = − +5664846191 +566524772352000 + +4001 +479001600 ln 2 = −4.20958 · 10−6 , +cf +10 = +68089272001 +83774850711552000 − +13817 +21794572800 ln 2 = 3.73334 · 10−7 , +(D.3) +We see that for values of the jet radius R < 1 the terms c6, c8 and c10 can be dropped. +For the gluon case the expansion of the function in numerical form is, +f(R, A) = − (1.0963 CA + 0.1768 TF nf) ln R + (0.6106 CA − 0.0310 TF nf) ++ (−0.5585 CA + 0.0221 TF nf) R2 ++ (0.0399 CA − 0.0004 TF nf) R4 + . . . , +(D.4) +whereas for the quark case we have +f(R, F) = − (1.0963 CA + 0.1768 TF nf) ln R + (0.6106 CA − 0.0310 TF nf) ++ (−0.8225 CF + 0.2639 CA + 0.0221 TF nf) R2 ++ (0.0625 CF − 0.02258 CA − 0.0004 TF nf) R4 + . . . . +(D.5) +– 43 – + +E +Renormalization Group Evolution +The evolution equation matching for a generic hard matching coefficient C has the form, +d +d ln µ ln C(Q2, µ) = +� +Γcusp(αs(µ)) ln Q2 +µ2 + γ(αs(µ)) +� +. +(E.1) +Following ref. [26] the solution to the evolution equation Eq. (E.1) is, +C(Q2, µ) = exp [2S(µh, µ) − aγ(µh, µ)] +�Q2 +µ2 +h +�−aΓ(µh,µ) +C(Q2, µh) , +(E.2) +ln C(Q2, µ) = 2S(µh, µ) − aγ(µh, µ) − aΓ(µh, µ) ln +�Q2 +µ2 +h +� ++ ln C(Q2, µh) , +(E.3) +where µh ∼ Q is a hard matching scale at which the Wilson coefficient C is calculated using +fixed-order perturbation theory. The Sudakov exponent S and the exponents aγ, aΓ are the +solutions to the auxiliary differential equations, +d +d ln µ S(ν, µ) = −Γcusp +� +αs(µ) +� +ln µ +ν , +(E.4) +d +d ln µ aΓ(ν, µ) = −Γcusp +� +αs(µ) +� +, +(E.5) +d +d ln µ aγ(ν, µ) = −γ +� +αs(µ) +� +. +(E.6) +with the boundary conditions S(ν, ν) = aΓ(ν, ν) = aγ(ν, ν) = 0 at µ = ν. Differentiating +Eq. (E.3) we recover Eq. (E.1). +The solutions to the evolution equation are conveniently expressed in terms of the running +coupling, +aΓ(ν, µ) = − +αs(µ) +� +αs(ν) +dα Γcusp(α) +β(α) +, +(E.7) +S(ν, µ) = − +αs(µ) +� +αs(ν) +dα Γcusp(α) +β(α) +α +� +αs(ν) +dα′ +β(α′) . +(E.8) +Substituting the values for the beta function coefficients in the MS scheme given in Appendix B.1 +and the values for cusp anomalous dimension given in Appendix B.2 into Eq. (E.7) we +obtain, +aΓ(µh, µ) = aΓ +0 + aΓ +1 + aΓ +2 + aΓ +3 , +(E.9) +– 44 – + +where the coefficients in the expansion are, +aΓ +0 = Γ0 ln(r) +2β0 +, +r = αs(µ)/αs(µh) , +(E.10) +aΓ +1 = αs(µh)(r − 1)(β0Γ1 − β1Γ0) +8πβ2 +0 +, +(E.11) +aΓ +2 = α2 +s(µh)(r2 − 1) +� +−β0β1Γ1 + β0(β0Γ2 − β2Γ0) + β2 +1Γ0 +� +64π2β3 +0 +, +(E.12) +aΓ +3 = −α3 +s(µh) +� +r3 − 1 +� +× +� +β2 +0(−β0Γ3 + β2Γ1 + β3Γ0) − β0β2 +1Γ1 + β0β1(β0Γ2 − 2β2Γ0) + β3 +1Γ0 +� +384π3β4 +0 +. +(E.13) +The solution for aγ follows from the one for aΓ by making the replacement Γk → γk. The +non-cusp anomalous dimensions γ are given in Appendix B.3. +Evaluating Eq. (E.8) to obtain the evolution for S we get, +S(µh, µ) = S0 + S1 + S2 . +(E.14) +with, +S0 = +1 +8β3 +0 +� +8πβ0Γ0(r + r(− ln(r)) − 1) +αs(µh)r ++ 2(r − 1)(β1Γ0 − β0Γ1) ++ ln(r)(2β0Γ1 + β1Γ0 ln(r) − 2β1Γ0) +� +, +(E.15) +S1 = −αs(µh) +32πβ4 +0 +� +2 ln(r) +� +−β0β1Γ1r + β0β2Γ0 + β2 +1Γ0(r − 1) +� ++ (r − 1) +� +−β0β1Γ1(r − 3) + β0(β0(r − 1)Γ2 − β2Γ0(r + 1)) + β2 +1Γ0(r − 1) +� +� +,(E.16) +S2 = α2 +s(µh) +256π2β5 +0 +� +2 ln(r) +� +β1r2 � +−β0β1Γ1 + β0(β0Γ2 − β2Γ0) + β2 +1Γ0 +� +− Γ0 +� +β2 +0β3 − 2β0β1β2 + β3 +1 +� � ++ (r − 1) +� +β2 +0(2(β0(r + 1)Γ3 − 2β2Γ1) − β3Γ0(r + 1)) + β0β2 +1Γ1(r + 5) ++ β0β1(β2Γ0(r + 5) − 3β0(r + 1)Γ2) − 4β3 +1Γ0 +�� +. +(E.17) +E.1 +Recovery of the double log formula +As we have seen S satisfies a RGE given by Eq. (E.4) with a solution given by Eq. (E.8). The +leading term in S0, Eq. (E.15) is +S0 ≈ +πΓ0 +β2 +0αs(µh) +� +1 + ln +�1 +r +� +− 1 +r +� +, +(E.18) +– 45 – + +where r = αs(µ)/αs(µh). In this form the presence of a double log is obscured. We can easily +recover the double log by retaining only the leading terms. The leading expression for r is +given by solving the equation for the beta function, +1 +r = 1 − αs(µh) +2π +β0 ln +�µh +µ +� +, +(E.19) +S0 ≈ +πΓ0 +β2 +0αs(µh) +�αs(µh) +2π +β0 ln +�µh +µ +� ++ ln +� +1 − αs(µh) +2π +β0 ln +�µh +µ +��� +. +(E.20) +Expanding for small αs(µh) ln(µh/µ) we get, +S(µh, µ) ≈ −Γ0 +2 +αS(µh) +4π +ln2 �µh +µ +� +. +(E.21) +This gives the expected log squared with a negative sign. +F +The hard function for the Drell-Yan process +The form factors of the vector current have been presented several places in the literature [79–84]. +The bare form factor is given as, +F q,bare(q2, µ2) = 1 + +�αbare +s +4π +� +(∆)ϵFq +1 + +�αbare +s +4π +�2 +(∆)2ϵFq +2 + O(α3 +s) , +(F.1) +where, +∆ = 4πe−γE +� +µ2 +−q2 − i0 +� +. +(F.2) +In the following we will drop 4πe−γE, so that all poles should be understood in the MS sense. +The values found for the bare coefficients are, +Fq +1 = CF +� +− 2 +ϵ2 − 3 +ϵ + ζ2 − 8 + ϵ +�3ζ2 +2 + 14ζ3 +3 +− 16 +� ++ ϵ2 +�47ζ2 +2 +20 ++ 4ζ2 + 7ζ3 − 32 +�� ++ O(ϵ3) , +(F.3) +Fq +2 = C2 +F +� +2 +ϵ4 + 6 +ϵ3 − 1 +ϵ2 +� +2ζ2 − 41 +2 +� +− 1 +ϵ +�64ζ3 +3 +− 221 +4 +� +− +� +13ζ2 +2 − 17ζ2 +2 ++ 58ζ3 − 1151 +8 +�� ++ CF CA +� +− 11 +6ϵ3 + 1 +ϵ2 +� +ζ2 − 83 +9 +� +− 1 +ϵ +�11ζ2 +6 +− 13ζ3 + 4129 +108 +� ++ +�44ζ2 +2 +5 +− 119ζ2 +9 ++ 467ζ3 +9 +− 89173 +648 +�� ++ CF nf +� +1 +3ϵ3 + 14 +9ϵ2 + 1 +ϵ +�ζ2 +3 + 353 +54 +� ++ +�14ζ2 +9 +− 26ζ3 +9 ++ 7541 +324 +�� ++ O(ϵ) . +(F.4) +– 46 – + +The renormalized form factor can then be written as, +F q(µ2, q2, ϵ) = 1 + +�αs(µ) +4π +� +F q +1 (µ2, q2, ϵ) + +�αs(µ) +4π +�2 +F q +2 (µ2, q2, ϵ) + O(α3 +s) . +(F.5) +where, +F q +1 (µ2, q2, ϵ) = ∆ϵFq +1 , +F q +2 (µ2, q2, ϵ) = ∆2ϵFq +2 − β0 +ϵ ∆ϵFq +1 . +(F.6) +In the full theory the matrix element between on-shell massless quark and gluon states, after +charge renormalization is given by F q(µ2, q2, ϵ). Charge renormalization has removed the UV +poles, but the renormalized form factor still contains IR poles. +The matrix element in the effective theory involves only scaleless, dimensionally regulated +integrals and hence is equal to zero. This vanishing can be interpreted as a cancellation between +ultra-violet and infrared poles: +1 +ϵIR +− +1 +ϵUV +. +(F.7) +After matching, the IR poles in the on-shell matrix element are effectively transformed into UV +poles and need to be renormalized as follows, +CV (αs(µ2), µ2, q2) = lim +ϵ→0 +� +ZV (ϵ, µ2q2) +�−1 F q(µ2, q2, ϵ) , +ln +� +CV (αs(µ2), µ2, q2) +� += ln +� +Fq(µ2, q2, ϵ) +� +− ln +� +ZV (ϵ, µ2, q2) +� +. +(F.8) +The renormalization constant, ZV contains only pure pole terms, +ln ZV (ϵ, µ2, q2) = +�αs +4π +� +� +− ΓF +0 +2ϵ2 + 1 +2ϵ +� +ΓF +0 L + 2γq +0 +�� ++ +�αs +4π +�2 +� +3ΓF +0 β0 +8ϵ3 +− 1 +ϵ2 +�ΓF +0 β0 +4 +L − CF +� +CA(16 +9 + ζ2 +� ++ 4 +9nf) +� ++ 1 +4ϵ +� +ΓF +1 L + 2γq +1 +�� +, (F.9) +where L = ln((−q2 − i0)/µ2). +The matching coefficients have a perturbative expansion in terms of the renormalized cou- +pling, +CV (αs(µ2), µ2, q2) = 1 + +∞ +� +n=1 +�αs(µ2) +4π +�n +CV +n (µ2, q2). +(F.10) +The matching coefficients, which are known to two loop order [85, 86] (and beyond [84]) for +Drell-Yan production, can be obtained from Eq. (F.8): +CV +1 = CF +� +− L2 + 3L − 8 + ζ2 +� +, +(F.11) +– 47 – + +CV +2 = C2 +F +�1 +2L4 − 3L3 + +�25 +2 − ζ2 +� +L2 + +� +− 45 +2 + 24ζ3 − 9ζ2 +� +L ++ 255 +8 +− 30ζ3 + 21ζ2 − 83 +10ζ2 +2 +� ++CF CA +�11 +9 L3 + +� +− 233 +18 + 2ζ2 +� +L2 + +�2545 +54 +− 26ζ3 + 22 +3 ζ2 +� +L +− 51157 +648 ++ 313 +9 ζ3 − 337 +18 ζ2 + 44 +5 ζ2 +2 +� ++ CF nf +� +− 2 +9L3 + 19 +9 L2 + +� +− 209 +27 − 4 +3ζ2 +� +L + 4085 +324 + 2 +9ζ3 + 23 +9 ζ2 +� +, +(F.12) +where L = ln((−q2 − i0)/µ2). CV satisfies the renormalization group equation, +d +d ln µ ln[CV (αs(µ2), µ2, q2)] = ΓF +cusp(µ) ln +�−q2 − i0 +µ2 +� ++ 2γq(µ) , +(F.13) +with the anomalous dimensions as given in Appendix B.2 and Appendix B.3. +The derivation of the hard function for boson pair processes has been described in Ref. [87]. +G +The hard function for Higgs production +G.1 +Implementation of one-step procedure +The one-step procedure [1, 13] is based on the observation that the ratio mt/mH is not large. +For an on-shell Higgs boson the parameter, m2 +H/m2 +t ≈ 1 +2 whereas αs ln(m2 +t /m2 +H) ≈ 0.65αs, +indicating that power corrections should be more important than resumming logarithms. The +matching is performed at a scale µh by integrating out the top quark and all gluons and light +quarks with off-shellness above µh. +The hard Wilson coefficient so defined satisfies the RGE, +µ d +dµ ln CH(m2 +t , q2, µ2) = ΓA +cusp(αs(µ)) ln −q2 − i0 +µ2 ++ 2γg[αs(µ)] , +(G.1) +where Γcusp and γg are given in Eqs. (B.5) and (B.11). As a consequence of Eq. (G.1) the +Wilson coefficient has the following structure, +CH(m2 +t , q2, µ2 +h) = αs(µh)F H +0 +� q2 +4m2 +t +�� +1 + αs(µh) +4π +� +CH +1 +�−q2 − i0 +µ2 +h +� ++ F H +1 +� q2 +4m2 +t +�� ++ +� +αs(µh) +(4π) +�2� +CH +2 +�−q2 − i0 +µ2 +h +, q2 +4m2 +t +� ++ F H +2 +� q2 +4m2 +t +��� +, +(G.2) +The finite terms can be derived from Ref. [88], +F H +0 (z) = 3 +2z − 3 +2z +���1 − 1 +z +��� +� +arcsin2(√z) , +0 < z ≤ 1 , +ln2[−i(√z + √z − 1)] , +z > 1 , +(G.3) +– 48 – + +≈ 1 + 7z +30 + 2z2 +21 + 26z3 +525 + 512z4 +17325 + O(z5), +z < 1 . +(G.4) +For the values of mt and mH in Table 2, +|F H +0 (z0)|2 = 1.0653 , +z0 = m2 +H +4m2 +t +. +(G.5) +The coefficients CH +1 and CH +2 are fixed by the Eq. (G.1). +CH +1 (L) = CA +� +−L2 + π2 +6 +� +, +(G.6) +CH +2 (L, z) = 1 +2C2 +AL4 + 1 +3CAβ0L3 + CA +�� +−4 +3 + π2 +6 +� +CA − 5 +3β0 − F1(z) +� +L2 ++ +��59 +9 − 2ζ3 +� +C2 +A + +�19 +9 − π2 +3 +� +CAβ0 − F1(z)β0 +� +L . +(G.7) +where z = q2/4/m2 +t and L = ln[(−q2 − i0)/µ2 +h]. +The full analytic mt dependence of the virtual two-loop corrections to gg → H in terms of +harmonic polylogarithms were obtained in Refs. [89–91]. For our purposes the results expanded +in m2 +H/m2 +t from Refs. [88, 92, 93] will be sufficient. +The functions F H +1 (z), F H +2 (z) which, +together with F H +0 (z) in Eq. (G.4) encode the mt dependence of the hard Wilson coefficient in +Eq. (G.2). Following the procedure described in Appendix F they are easily extracted from +Ref. [88], +F H +1 (z) = +� +5 − 38 +45 z − 1289 +4725 z2 − 155 +1134 z3 − 5385047 +65488500 z4� +CA ++ +� +−3 + 307 +90 z + 25813 +18900 z2 + 3055907 +3969000 z3 + 659504801 +1309770000 z4� +CF + O(z5) +(G.8) +F H +2 (z) = +� +7C2 +A + 11CACF − 6CF β0 +� +ln(−4z − i0) + +� +−419 +27 + 7π2 +6 ++ π4 +72 − 44ζ3 +� +C2 +A ++ +� +−217 +2 +− π2 +2 + 44ζ3 +� +CACF + +�2255 +108 + 5π2 +12 + 23ζ3 +3 +� +CAβ0 − 5 +6CATF ++ 27 +2 C2 +F + +�41 +2 − 12ζ3 +� +CF β0 − 4 +3CF TF ++ z +� +C2 +A +�11723 +384 ζ3 − 404063 +14400 − 223 +108 ln(−4z − i0) − 19 +135π2� ++ CF CA +�2297 +16 ζ3 − 1099453 +8100 +− 242 +135 ln(−4z − i0) − 953 +540π2 + 28 +15π2 ln 2 +� ++ C2 +F +�13321 +96 +ζ3 − 36803 +240 ++ 7 +3π2 − 56 +15π2 ln 2 +� ++ CF +�77 +12ζ3 − 4393 +405 − 7337 +2700β0 + 39 +10 ln(−4z − i0)β0 + 28 +45π2 + 7 +15π2β0 +� ++ CA +� 77 +384ζ3 − 64097 +129600 − 269 +75 β0 + 2 +15 ln(−4z − i0) − 31 +180 ln(−4z − i0)β0 +�� ++ z2� +C2 +A +�110251 +9216 ζ3 − 3084463261 +254016000 − 2869 +4536 ln(−4z − i0) − 1289 +28350π2� +– 49 – + ++ CF CA +�2997917 +23040 ζ3 − 55535378557 +381024000 +− 18337 +28350 ln(−4z − i0) − 128447 +113400π2 + 1714 +1575π2 ln 2 +� ++ C2 +F +�36173 +192 ζ3 − 95081911 +453600 ++ 857 +630π2 − 3428 +1575π2 ln 2 +� ++ CA +� 265053121 +1524096000 − 16177 +92160ζ3 − 45617 +47250β0 + 16 +315 ln(−4z − i0) − 623 +5400 ln(−4z − i0)β0 +� ++ CF +�21973 +7680 ζ3 − 8108339 +1555200 − 509813 +3969000β0 − 8 +15 ln(−4z − i0) + 29147 +18900 ln(−4z − i0)β0 ++ 1714 +4725π2 + 857 +3150π2β0 +�� ++ O(z3) . +(G.9) +We can assess the quality of the expansion in z by numerical evaluation, +CH(m2 +t , q2, q2) = αs(q)F0(z) +� +1 + 15.9348αs +4π(1 + 0.0158(8z) + .00098312(8z)2) ++ 97.0371 +�αs +4π +�2 +(1 + 0.1883(8z) + 0.0120(8z)2) ++ 143.466 +�αs +4π +�2 ln(−8z − i0) +π +(1 + 0.0288(8z) + 0.001462(8z)2) +� +. +(G.10) +In the vicinity of the Higgs boson pole (8z ≈ 1) subsequent terms in the z expansion are +expected to contribute below the percent level. +G.2 +Implementation of the two-step procedure +In the two-step procedure of Refs. [59–62] one first integrates out the top quark at a scale +µt ≊ mt and subsequently matches from the QCD effective Lagrangian onto SCET at µh ≊ mH. +Running between µh and µt allows one to sum logarithms of mt/mH, but one neglects power +of mH/mt. +G.2.1 +Ct(m2 +t , µ2 +t ) +For a heavy top quark the effective Lagrangian for the production of a top quark is given +by, +Leff = Ct(m2 +t , µ2 +t ) H +v +αs(µ2 +t ) +12π +Gµν aGµν +a , +(G.11) +where v ≈ 246 GeV is the Higgs boson vacuum expectation value. The hard matching scale +µt at which the Wilson coefficient can be computed perturbatively is of order mt. The short +distance coefficient Ct(m2 +t , µ2) obeys the RGE, +d +d ln µCt(m2 +t , µ2) = γt(αs) Ct(m2 +t , µ2), +γt(αs) = α2 +s +d +dαs +�β(αs) +α2s +� +. +(G.12) +The expressions for the short-distance coefficient Ct(m2 +t , µ2 +t ) at NNLO is, +Ct(m2 +t , µ2 +t ) = 1 + αs(µt) +4π +Ct +1 + +�αs(µt) +4π +�2 +Ct +2(m2 +t , µ2 +t ) + . . . , +(G.13) +where (c.f. Eq. (12) of Ref. [61]), +Ct +1 = 5CA − 3CF +– 50 – + +Ct +2(m2 +t , µ2 +t ) = 27 +2 C2 +F + +� +11 ln m2 +t +µ2 +t +− 100 +3 +� +CF CA − +� +7 ln m2 +t +µ2 +t +− 1063 +36 +� +C2 +A +−4 +3CF TF − 5 +6CATF − +� +8 ln m2 +t +µ2 +t ++ 5 +� +CF TF nf − 47 +9 CATF nf . (G.14) +The evolution of these coefficients to the resummation scale µ is described in Appendix A of +Ref. [3]. The solution to the evolution equation Eq. (G.12) for Ct at scale µ is, +Ct(m2 +t , µ2) = β(αs(µ)) +α2s(µ) +α2 +s(µt) +β(αs(µt)) Ct(m2 +t , µ2 +t ) . +(G.15) +The result at NNLO for the square of the coefficient function is, +� +Ct(m2 +t , µ2) +�2 = 1 + +�αs +4π +�� +2Ct +1 + 2(rt − 1)β1 +β0 +� ++ +�αs +4π +�2� +(Ct +1)2 + 2Ct +2(m2 +t , µ2 +t ) + (2β2β0 + β2 +1) +β2 +0 +(rt − 1)2 ++ 2(2β2β0 + 2β1β0Ct +1 − β2 +1) +β2 +0 +(rt − 1) +� +, +(G.16) +where rt = αs(µ)/αs(µt). This extends the NLO result in Eq. (2) of Ref. [3]. +G.2.2 +CS(−q2, µh) +CS is the Wilson coefficient matching the two gluon operator in Eq. (G.11) to an operator +in SCET in which all the hard modes have been integrated out. The result for the matching +coefficient CS from Eqs.(16) and (17) of Ref. [61]. It is given by, +CS(−q2, µ2 +h) = 1 + +∞ +� +n=1 +CS +n (L) +�αs(µ2 +h) +4π +�n +. +(G.17) +The coefficient CS obeys the renormalization equation, +d +d ln µ CS(−q2 − iϵ, µ2) = +� +ΓA +cusp(αs) ln −q2 − iϵ +µ2 ++ γS(αs) +� +CS(−q2 − iϵ, µ2) , +(G.18) +with L = ln(−q2 − i0)/µ2 +h and γS is given in Eq (B.20). +The logarithmic terms are determined by Eq. (G.18). The full results for the one- and two-loop +coefficients are, +CS +1 = CA +� +− L2 + π2 +6 +� +, +(G.19) +CS +2 = C2 +A +�L4 +2 + 11 +9 L3 + +� +− 67 +9 + π2 +6 +� +L2 + +�80 +27 − 11π2 +9 +− 2ζ3 +� +L ++ 5105 +162 + 67π2 +36 ++ π4 +72 − 143 +9 ζ3 +� ++ CF TF nf +� +4L − 67 +3 + 16ζ3 +� +– 51 – + ++ CATF nf +� +− 4 +9 L3 + 20 +9 L2 + +�104 +27 + 4π2 +9 +� +L − 1832 +81 +− 5π2 +9 +− 92 +9 ζ3 +� +. +(G.20) +The full result for the renormalization group invariant hard function in the two-step scheme +is, +¯H(mt, mH, pveto +T +) = +� αs(µ) +αs(pveto +T +) +�2 +(Ct(m2 +t , µ))2 ��CS(−m2 +H, µ) +��2 +× +� mH +pveto +T +�−2Fgg(pveto +T +,µ) +e2hA(pveto +T +,µ) . +(G.21) +The µ-independence of this hard function can be used to constrain γS, +d +d ln µ +¯H(mt, mH, pveto +T +) = 0 . +(G.22) +Using Eqs. (B.1,G.12,G.18,2.13,C.1) we can derive the relation between the collinear anomalous +dimensions, +2γg(αs) = γt(αs) + γS(αs) + β(αs)/αs . +(G.23) +This relation could be cast in a more transparent form by noting that the quantity (αsCS) +obeys a similar evolution equation to Eq. (G.18), +d +d ln µ +� +αs(µ)CS(−m2 +H − iϵ, µ2) +� += +αs(µ) +� +ΓA +cusp(αs) ln −m2 +H − iϵ +µ2 ++ γS(αs) +� +CS(−m2 +H − iϵ, µ2) + β(αs)CS(−m2 +H − iϵ, µ2) += +� +ΓA +cusp(αs) ln −m2 +H − iϵ +µ2 ++ γS′(αs) +� � +αs(µ)CS(−m2 +H − iϵ, µ2) +� +, +(G.24) +but with anomalous dimension γS′(αs) = γS(αs) + β(αs)/αs. We then have the relation +2γg(αs) = γt(αs) + γS′(αs). This indicates that after the second matching, the evolution +down to a lower scale satisfies the same renormalization equation in both the one-step and the +two-step schemes. +G.3 +Assessment of the two schemes for the Higgs hard function +The two schemes for the calculation of the hard function have application in jet veto resum- +mation but also in the resummation of the Higgs boson transverse momentum. A complete +discussion of the error budget for Higgs boson production including scale dependence, parton +distribution dependence, the influence of loops of b-quarks and electroweak corrections is +beyond the scope of this paper. Here we shall simply compare and contrast the one-step and +the two-step scheme, in the Higgs on shell region where m2 +H ≈ m2 +t /2. +It is easy to check the internal consistency of the two schemes in the limit where we drop +terms of order q2/(4m2 +t ). Setting z = 0 in Eq. (G.2) and evaluating all coefficient functions at +– 52 – + +a common scale µ, we have that, +αs(µ) Ct(m2 +t , µ2) CS(−q2, µ2) = CH(m2 +t , q2, µ2)z=0 + O(α4 +s) . +(G.25) +We can test this equivalence numerically. We start by fixing µ2 = q2 and consider the quantities +that enter the calculation of the cross-section, i.e. the square of the absolute values. In the +two-step scheme we have, +|Ct(m2 +t , q2)|2 = 1 + 0.1957 + 0.0204 , +|Cs(−q2, q2)|2 = 1 + 0.6146 + 0.2155 , +(G.26) +where the second and third terms represent the O(αs) and O(α2 +s) terms respectively, evaluated +using αs(q2) = 0.1118. In the one-step case we get, +|CH +z=0(m2 +t , q2, q2)/αs(q)|2 = 1 + 0.8104 + 0.3563 . +(G.27) +Performing a strict fixed-order truncation of the product of the two-step result we have, +� +|Ct(m2 +t , q2)|2|Cs(−q2, q2)|2� +expanded = 1 + 0.8104 + 0.3563 , +(G.28) +which is in perfect agreement with the one-step case. This indicates that the numerical +implementation of the two procedures is correct. If we instead evaluate the product after the +individual expansions have been performed, a choice of equal formal accuracy, we have, +|Ct(m2 +t , q2)|2 +expanded |Cs(−q2, q2)|2 +expanded = 1 + 0.9306 + 0.2953 . +(G.29) +This results in a significant difference. We therefore work with with the strict fixed-order +truncation throughout this paper. +We now restore the z-dependence in F H +1 +and F H +2 +in Eq. (G.2), but still keep z = 0 in the +overall factor F H +0 (z). We then find that the ratio of the one-step to the two-step becomes +1.0028 at NLO and 1.0053 at NNLO, i.e. these corrections are very small. Now we allow the +matching scale for the top quark, µt to take its natural value, µt = mt and find one/two-step +ratios of 1.0054 at NLO and 1.0073 at NNLO, again a small effect. Finally, we reinstate the +hard evolution down to the resummation scale and find that the ratio of the one-step to the +two-step (at pveto +T += 25 GeV) is 1.0177 at NLO and 1.0125 at NNLO. The cumulative effect at +this point is noticeable but still small. However, we note that we have so far kept z = 0 in +the overall factor F H +0 (z). The one-step procedure is recovered by re-instating F H +0 (z). This +implies that, in order to obtain the level of agreement quoted above between the two schemes, +the overall factor of F H +0 (z) must also be applied to give a modified version of the two-step +scheme. Neglecting this step would result in a significant difference, since |F H +0 (z)|2 = 1.0653 +see Eq.(G.5). +Our overall conclusion on the two schemes is in line with the known result that Higgs boson +production has substantial corrections. Accounting for the most important mass effects by +– 53 – + +rescaling the two-step result by the exact result at leading order, the one-step procedure +gives a larger result than the two-step procedure for pveto +T += 25 GeV at the level of 1.3%. +Any substantial difference between the two methods beyond this level is most likely due to +uncontrolled higher order effects. +References +[1] C.F. Berger, C. Marcantonini, I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, Higgs +Production with a Central Jet Veto at NNLL+NNLO, JHEP 04 (2011) 092 [1012.4480]. +[2] I.W. Stewart, F.J. Tackmann and W.J. Waalewijn, N-Jettiness: An Inclusive Event Shape to +Veto Jets, Phys. Rev. Lett. 105 (2010) 092002 [1004.2489]. +[3] T. Becher and M. Neubert, Factorization and NNLL Resummation for Higgs Production with a +Jet Veto, JHEP 07 (2012) 108 [1205.3806]. +[4] A. Banfi, G.P. Salam and G. Zanderighi, NLL+NNLO predictions for jet-veto efficiencies in +Higgs-boson and Drell-Yan production, JHEP 06 (2012) 159 [1203.5773]. +[5] A. Banfi, P.F. Monni, G.P. Salam and G. Zanderighi, Higgs and Z-boson production with a jet +veto, Phys. Rev. Lett. 109 (2012) 202001 [1206.4998]. +[6] S. Kallweit, E. Re, L. Rottoli and M. Wiesemann, Accurate single- and double-differential +resummation of colour-singlet processes with MATRIX+RADISH: W +W − production at the +LHC, JHEP 12 (2020) 147 [2004.07720]. +[7] E. Re, L. Rottoli and P. Torrielli, Fiducial Higgs and Drell-Yan distributions at N3LL′+NNLO +with RadISH, 2104.07509. +[8] T. Becher, R. Frederix, M. Neubert and L. Rothen, Automated NNLL + NLO resummation for +jet-veto cross sections, Eur. Phys. J. C 75 (2015) 154 [1412.8408]. +[9] A. Banfi, P.F. Monni, G.P. Salam and G. Zanderighi, “JetVHeto.” +https://jetvheto.hepforge.org/, 2016. +[10] L. Arpino, A. Banfi, S. Jäger and N. Kauer, “MCFM-RE.” +https://github.com/lcarpino/MCFM-RE, 2019. +[11] L.R. Stefan Kallweit, Emanuele Re and M. Wiesemann, “MCFM-RE.” +https://matrix.hepforge.org/matrix+radish.html, 2020. +[12] T. Becher, M. Neubert and L. Rothen, Factorization and N 3LLp+NNLO predictions for the +Higgs cross section with a jet veto, JHEP 10 (2013) 125 [1307.0025]. +[13] I.W. Stewart, F.J. Tackmann, J.R. Walsh and S. Zuberi, Jet pT resummation in Higgs production +at NNLL′ + NNLO, Phys. Rev. D 89 (2014) 054001 [1307.1808]. +[14] A. Banfi, F. Caola, F.A. Dreyer, P.F. Monni, G.P. Salam, G. Zanderighi et al., Jet-vetoed Higgs +cross section in gluon fusion at N3LO+NNLL with small-R resummation, JHEP 04 (2016) 049 +[1511.02886]. +[15] S. Dawson, P. Jaiswal, Y. Li, H. Ramani and M. Zeng, Resummation of jet veto logarithms at +N3LLa + NNLO for W +W − production at the LHC, Phys. Rev. D 94 (2016) 114014 +[1606.01034]. +[16] L. Arpino, A. Banfi, S. Jäger and N. Kauer, BSM WW production with a jet veto, JHEP 08 +(2019) 076 [1905.06646]. +– 54 – + +[17] Y. Wang, C.S. Li and Z.L. Liu, Resummation prediction on gauge boson pair production with a +jet veto, Phys. Rev. D 93 (2016) 094020 [1504.00509]. +[18] G.P. Salam, Towards Jetography, Eur. Phys. J. C 67 (2010) 637 [0906.1833]. +[19] M. Cacciari, G.P. Salam and G. Soyez, The anti-kt jet clustering algorithm, JHEP 04 (2008) 063 +[0802.1189]. +[20] Y.L. Dokshitzer, G.D. Leder, S. Moretti and B.R. Webber, Better jet clustering algorithms, +JHEP 08 (1997) 001 [hep-ph/9707323]. +[21] M. Wobisch and T. Wengler, Hadronization corrections to jet cross-sections in deep inelastic +scattering, in Workshop on Monte Carlo Generators for HERA Physics (Plenary Starting +Meeting), pp. 270–279, 4, 1998 [hep-ph/9907280]. +[22] S. Catani, Y.L. Dokshitzer, M.H. Seymour and B.R. Webber, Longitudinally invariant Kt +clustering algorithms for hadron hadron collisions, Nucl. Phys. B 406 (1993) 187. +[23] S.D. Ellis and D.E. Soper, Successive combination jet algorithm for hadron collisions, Phys. Rev. +D 48 (1993) 3160 [hep-ph/9305266]. +[24] S. Abreu, J.R. Gaunt, P.F. Monni, L. Rottoli and R. Szafron, Quark and gluon two-loop beam +functions for leading-jet pT and slicing at NNLO, 2207.07037. +[25] S. Abreu, J.R. Gaunt, P.F. Monni and R. Szafron, The analytic two-loop soft function for +leading-jet pT , 2204.02987. +[26] T. Becher, M. Neubert and B.D. Pecjak, Factorization and Momentum-Space Resummation in +Deep-Inelastic Scattering, JHEP 01 (2007) 076 [hep-ph/0607228]. +[27] Y. Li, D. Neill and H.X. Zhu, An exponential regulator for rapidity divergences, Nucl. Phys. B +960 (2020) 115193 [1604.00392]. +[28] A.A. Vladimirov, Correspondence between Soft and Rapidity Anomalous Dimensions, Phys. Rev. +Lett. 118 (2017) 062001 [1610.05791]. +[29] Y. Li and H.X. Zhu, Bootstrapping Rapidity Anomalous Dimensions for Transverse-Momentum +Resummation, Phys. Rev. Lett. 118 (2017) 022004 [1604.01404]. +[30] G. Billis, M.A. Ebert, J.K.L. Michel and F.J. Tackmann, A toolbox for qT and 0-jettiness +subtractions at N3LO, Eur. Phys. J. Plus 136 (2021) 214 [1909.00811]. +[31] T. Becher and T. Neumann, Fiducial qT resummation of color-singlet processes at N3LL+NNLO, +JHEP 03 (2021) 199 [2009.11437]. +[32] J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A Formalism for the Systematic Treatment of +Rapidity Logarithms in Quantum Field Theory, JHEP 05 (2012) 084 [1202.0814]. +[33] J.-y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, The Rapidity Renormalization Group, Phys. Rev. +Lett. 108 (2012) 151601 [1104.0881]. +[34] S. Alioli and J.R. Walsh, Jet Veto Clustering Logarithms Beyond Leading Order, JHEP 03 (2014) +119 [1311.5234]. +[35] M. Dasgupta, F. Dreyer, G.P. Salam and G. Soyez, Small-radius jets to all orders in QCD, JHEP +04 (2015) 039 [1411.5182]. +[36] NNPDF collaboration, Parton distributions from high-precision collider data, Eur. Phys. J. C 77 +(2017) 663 [1706.00428]. +[37] A. Banfi, P.F. Monni and G. Zanderighi, Quark masses in Higgs production with a jet veto, +JHEP 01 (2014) 097 [1308.4634]. +– 55 – + +[38] CMS collaboration, Measurement of differential cross sections for the production of a Z boson in +association with jets in proton-proton collisions at √s = 13 TeV, 2205.02872. +[39] CMS collaboration, W+W− boson pair production in proton-proton collisions at √s = 13 TeV, +Phys. Rev. D 102 (2020) 092001 [2009.00119]. +[40] CMS collaboration, Measurements of pp → ZZ production cross sections and constraints on +anomalous triple gauge couplings at √s = 13 TeV, Eur. Phys. J. C 81 (2021) 200 [2009.01186]. +[41] P. Jaiswal and T. Okui, Reemergence of rapidity-scale uncertainty in soft-collinear effective +theory, Phys. Rev. D 92 (2015) 074035 [1506.07529]. +[42] J.K.L. Michel, P. Pietrulewicz and F.J. Tackmann, Jet Veto Resummation with Jet Rapidity Cuts, +JHEP 04 (2019) 142 [1810.12911]. +[43] ATLAS collaboration, Measurement of differential cross sections and W +/W − cross-section +ratios for W boson production in association with jets at √s = 8 TeV with the ATLAS detector, +JHEP 05 (2018) 077 [1711.03296]. +[44] ATLAS collaboration, Measurement of W ±Z production cross sections and gauge boson +polarisation in pp collisions at √s = 13 TeV with the ATLAS detector, Eur. Phys. J. C 79 (2019) +535 [1902.05759]. +[45] CMS collaboration, Measurement of the inclusive and differential WZ production cross sections, +polarization angles, and triple gauge couplings in pp collisions at √s = 13 TeV, JHEP 07 (2022) +032 [2110.11231]. +[46] A. Banfi, G.P. Salam and G. Zanderighi, Principles of general final-state resummation and +automated implementation, JHEP 03 (2005) 073 [hep-ph/0407286]. +[47] P.F. Monni, L. Rottoli and P. Torrielli, Higgs transverse momentum with a jet veto: a +double-differential resummation, Phys. Rev. Lett. 124 (2020) 252001 [1909.04704]. +[48] W. Bizon, P.F. Monni, E. Re, L. Rottoli and P. Torrielli, Momentum-space resummation for +transverse observables and the Higgs p⊥ at N3LL+NNLO, JHEP 02 (2018) 108 [1705.09127]. +[49] P.F. Monni, E. Re and P. Torrielli, Higgs Transverse-Momentum Resummation in Direct Space, +Phys. Rev. Lett. 116 (2016) 242001 [1604.02191]. +[50] T. Becher, M. Neubert and D. Wilhelm, Electroweak Gauge-Boson Production at Small qT : +Infrared Safety from the Collinear Anomaly, JHEP 02 (2012) 124 [1109.6027]. +[51] V. Ahrens, T. Becher, M. Neubert and L.L. Yang, Origin of the Large Perturbative Corrections to +Higgs Production at Hadron Colliders, Phys. Rev. D 79 (2009) 033013 [0808.3008]. +[52] ATLAS collaboration, Measurement of the W +W − production cross section in pp collisions at a +centre-of -mass energy of √s = 13 TeV with the ATLAS experiment, Phys. Lett. B 773 (2017) +354 [1702.04519]. +[53] ATLAS collaboration, Measurement of fiducial and differential W +W − production cross-sections +at √s = 13 TeV with the ATLAS detector, Eur. Phys. J. C 79 (2019) 884 [1905.04242]. +[54] CMS collaboration, Search for anomalous triple gauge couplings in WW and WZ production in +lepton + jet ev ents in proton-proton collisions at √s = 13 TeV, JHEP 12 (2019) 062 +[1907.08354]. +[55] J.M. Campbell, R.K. Ellis, T. Neumann and S. Seth, Transverse momentum resummation at +N3LL+NNLO for diboson processes, 2210.10724. +[56] R. Bonciani, V. Del Duca, H. Frellesvig, M. Hidding, V. Hirschi, F. Moriello et al., +– 56 – + +Next-to-leading-order QCD Corrections to Higgs Production in association with a Jet, +2206.10490. +[57] M. Czakon, R.V. Harlander, J. Klappert and M. Niggetiedt, Exact Top-Quark Mass Dependence +in Hadronic Higgs Production, Phys. Rev. Lett. 127 (2021) 162002 [2105.04436]. +[58] T. Neumann and M. Wiesemann, Finite top-mass effects in gluon-induced Higgs production with +a jet-veto at NNLO, JHEP 11 (2014) 150 [1408.6836]. +[59] A. Idilbi, X.-d. Ji and F. Yuan, Transverse momentum distribution through soft-gluon +resummation in effective field theory, Phys. Lett. B 625 (2005) 253 [hep-ph/0507196]. +[60] A. Idilbi, X.-d. Ji, J.-P. Ma and F. Yuan, Threshold resummation for Higgs production in effective +field theory, Phys. Rev. D 73 (2006) 077501 [hep-ph/0509294]. +[61] V. Ahrens, T. Becher, M. Neubert and L.L. Yang, Renormalization-Group Improved Prediction +for Higgs Production at Hadron Colliders, Eur. Phys. J. C 62 (2009) 333 [0809.4283]. +[62] S. Mantry and F. Petriello, Factorization and Resummation of Higgs Boson Differential +Distributions in Soft-Collinear Effective Theory, Phys. Rev. D 81 (2010) 093007 [0911.4135]. +[63] G. Bell, K. Brune, G. Das and M. Wald, The NNLO quark beam function for jet-veto +resummation, 2207.05578. +[64] M.-X. Luo, X. Wang, X. Xu, L.L. Yang, T.-Z. Yang and H.X. Zhu, Transverse Parton +Distribution and Fragmentation Functions at NNLO: the Quark Case, JHEP 10 (2019) 083 +[1908.03831]. +[65] M.-X. Luo, T.-Z. Yang, H.X. Zhu and Y.J. Zhu, Transverse Parton Distribution and +Fragmentation Functions at NNLO: the Gluon Case, JHEP 01 (2020) 040 [1909.13820]. +[66] T. Becher and M. Neubert, Drell-Yan Production at Small qT , Transverse Parton Distributions +and the Collinear Anomaly, Eur. Phys. J. C 71 (2011) 1665 [1007.4005]. +[67] O.V. Tarasov, A.A. Vladimirov and A.Y. Zharkov, The Gell-Mann-Low Function of QCD in the +Three Loop Approximation, Phys. Lett. B 93 (1980) 429. +[68] S.A. Larin and J.A.M. Vermaseren, The Three loop QCD Beta function and anomalous +dimensions, Phys. Lett. B 303 (1993) 334 [hep-ph/9302208]. +[69] T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The Four loop beta function in quantum +chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390]. +[70] T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren, Group theory factors for Feynman +diagrams, Int. J. Mod. Phys. A 14 (1999) 41 [hep-ph/9802376]. +[71] J.M. Henn, G.P. Korchemsky and B. Mistlberger, The full four-loop cusp anomalous dimension +in N = 4 super Yang-Mills and QCD, JHEP 04 (2020) 018 [1911.10174]. +[72] A. von Manteuffel, E. Panzer and R.M. Schabinger, Cusp and collinear anomalous dimensions in +four-loop QCD from form factors, Phys. Rev. Lett. 124 (2020) 162001 [2002.04617]. +[73] T. Becher, A. Broggio and A. Ferroglia, Introduction to Soft-Collinear Effective Theory, vol. 896, +Springer (2015), 10.1007/978-3-319-14848-9, [1410.1892]. +[74] T. Becher and M. Neubert, On the Structure of Infrared Singularities of Gauge-Theory +Amplitudes, JHEP 06 (2009) 081 [0903.1126]. +[75] G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) +298. +– 57 – + +[76] G. Curci, W. Furmanski and R. Petronzio, Evolution of Parton Densities Beyond Leading Order: +The Nonsinglet Case, Nucl. Phys. B 175 (1980) 27. +[77] W. Furmanski and R. Petronzio, Singlet Parton Densities Beyond Leading Order, Phys. Lett. B +97 (1980) 437. +[78] R.K. Ellis, W.J. Stirling and B.R. Webber, QCD and collider physics, vol. 8, Cambridge +University Press (2, 2011), 10.1017/CBO9780511628788. +[79] G. Kramer and B. Lampe, Two Jet Cross-Section in e+ e- Annihilation, Z. Phys. C 34 (1987) +497. +[80] T. Matsuura and W.L. van Neerven, Second Order Logarithmic Corrections to the Drell-Yan +Cross-section, Z. Phys. C 38 (1988) 623. +[81] T. Matsuura, S.C. van der Marck and W.L. van Neerven, The Calculation of the Second Order +Soft and Virtual Contributions to the Drell-Yan Cross-Section, Nucl. Phys. B 319 (1989) 570. +[82] S. Moch, J.A.M. Vermaseren and A. Vogt, Three-loop results for quark and gluon form-factors, +Phys. Lett. B 625 (2005) 245 [hep-ph/0508055]. +[83] S. Moch, J.A.M. Vermaseren and A. Vogt, The Quark form-factor at higher orders, JHEP 08 +(2005) 049 [hep-ph/0507039]. +[84] T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark +and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [1004.3653]. +[85] A. Idilbi and X.-d. Ji, Threshold resummation for Drell-Yan process in soft-collinear effective +theory, Phys. Rev. D 72 (2005) 054016 [hep-ph/0501006]. +[86] A. Idilbi, X.-d. Ji and F. Yuan, Resummation of threshold logarithms in effective field theory for +DIS, Drell-Yan and Higgs production, Nucl. Phys. B 753 (2006) 42 [hep-ph/0605068]. +[87] J.M. Campbell, R.K. Ellis and S. Seth, Non-local slicing approaches for NNLO QCD in MCFM, +JHEP 06 (2022) 002 [2202.07738]. +[88] J. Davies, F. Herren and M. Steinhauser, Top Quark Mass Effects in Higgs Boson Production at +Four-Loop Order: Virtual Corrections, Phys. Rev. Lett. 124 (2020) 112002 [1911.10214]. +[89] M. Spira, A. Djouadi, D. Graudenz and P.M. Zerwas, Higgs boson production at the LHC, Nucl. +Phys. B 453 (1995) 17 [hep-ph/9504378]. +[90] R. Harlander and P. Kant, Higgs production and decay: Analytic results at next-to-leading order +QCD, JHEP 12 (2005) 015 [hep-ph/0509189]. +[91] C. Anastasiou, S. Beerli, S. Bucherer, A. Daleo and Z. Kunszt, Two-loop amplitudes and master +integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop, JHEP +01 (2007) 082 [hep-ph/0611236]. +[92] R.V. Harlander and K.J. Ozeren, Top mass effects in Higgs production at next-to-next-to-leading +order QCD: Virtual corrections, Phys. Lett. B 679 (2009) 467 [0907.2997]. +[93] A. Pak, M. Rogal and M. Steinhauser, Virtual three-loop corrections to Higgs boson production in +gluon fusion for finite top quark mass, Phys. Lett. B 679 (2009) 473 [0907.2998]. +– 58 – + diff --git a/3dFKT4oBgHgl3EQfQy0a/content/tmp_files/load_file.txt b/3dFKT4oBgHgl3EQfQy0a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3752c3c4d07ca5b00be316557aba2047d1f6658d --- /dev/null +++ b/3dFKT4oBgHgl3EQfQy0a/content/tmp_files/load_file.txt @@ -0,0 +1,2529 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf,len=2528 +page_content='Prepared for submission to JHEP FERMILAB-PUB-23-028-T, IPPP/23/05 Jet-veto resummation at N3LLp+NNLO in boson production processes John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Campbell,a R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Keith Ellis,b Tobias Neumann,c Satyajit Sethd aFermilab, PO Box 500, Batavia IL 60510-5011, USA bInstitute for Particle Physics Phenomenology, Durham University, Durham, DH1 3LE, UK cDepartment of Physics, Brookhaven National Laboratory, Upton, New York 11973, USA dPhysical Research Laboratory, Navrangpura, Ahmedabad - 380009, India E-mail: johnmc@fnal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='gov, keith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='ellis@durham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='uk, tneumann@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='gov, seth@prl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='in Abstract: Vetoing energetic jet activity is a crucial tool for suppressing backgrounds and enabling new physics searches at the LHC, but the introduction of a veto scale can introduce large logarithms that may need to be resummed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We present an implementation of jet-veto resummation for color-singlet processes at the level of N3LLp matched to fixed-order NNLO predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our public code MCFM allows for predictions of a single boson, such as Z/γ∗, W ± or H, or with a pair of vector bosons, such as W +W −, W ±Z or ZZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The implementation relies on recent calculations of the soft and beam functions in the presence of a jet veto over all rapidities, with jets defined using a sequential recombination algorithm with jet radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However one of the ingredients that is required to reach full N3LL accuracy is only known approximately, hence N3LLp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We describe in detail our formalism and compare with previous public codes that operate at the level of NNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our higher-order predictions improve significantly upon NNLL calculations by reducing theoretical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We demonstrate this by comparing our predictions with ATLAS and CMS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11768v1 [hep-ph] 27 Jan 2023 Contents 1 Introduction 1 2 Jet-veto factorization and resummation 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 The collinear anomaly coefficient and its approximations 5 3 Setup for phenomenology 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Input parameters 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 Uncertainty estimates at fixed order 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 Uncertainty estimates at the resummed and matched level 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 Effects of cuts on rapidity at fixed order 13 4 Comparison with JetVHeto 15 5 Phenomenological results 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Z and W production 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 W +W − production 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 W ±Z production 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 ZZ production 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 Higgs production 25 6 Conclusions 28 A Reduced beam functions 30 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Structure of the two-loop reduced beam function 31 B Definition of the beta function and anomalous dimensions 32 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Expansion of β-function 32 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 Cusp Anomalous Dimension 34 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 Non-cusp anomalous dimension 35 C Definitions for beam function ingredients 37 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Exponent h 37 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 One loop splitting functions 38 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 Two loop splitting functions 38 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 P (1) ⊗ P (1) and R(1) ⊗ P (1) 41 D Rapidity anomalous dimension 42 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 dveto 2 expansion 43 E Renormalization Group Evolution 44 – i – E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Recovery of the double log formula 45 F The hard function for the Drell-Yan process 46 G The hard function for Higgs production 48 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Implementation of one-step procedure 48 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 Implementation of the two-step procedure 50 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 Assessment of the two schemes for the Higgs hard function 52 1 Introduction Jet vetoing is a crucial technique in particle physics that is used primarily to suppress backgrounds in processes involving the production of W +W − final states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' directly or in Higgs decays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' By identifying and removing events that contain energetic hadronic jets (vetoing), the impact of the dominant top-quark pair production background is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The concrete jet-veto implementation depends on factors such as the jet algorithm and its parameters, as well as the kinematic selection cuts applied to the identified jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For LHC analyses, the most common jet vetoing scheme is to impose a maximum transverse momentum cut pveto T on anti-kT jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The jet veto scale pveto T can induce large logarithms if it is smaller than the hard process scale Q, which then mandates resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In this paper we describe a coherent implementation of jet veto resummation in processes involving the production of a color-singlet boson (W, Z/γ∗ and H bosons) or a pair of bosons (ZZ, W ±Z, and W +W −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our resummation operates at the level of N3LLp1 matched to fixed order NNLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We build on the pioneering work of previous studies, which have demonstrated the effectiveness of resummation methods for a jet veto [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' General purpose implementations include a numerical approach to resummation at NNLO+NNLL [6, 7] and an automated approach to jet veto studies at NLO+NNLL [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Publicly available codes operating at NNLL and addressing the same issue are, JetVHeto [9], the code MCFM-RE [10] which is derivative of both MCFM and JetVHeto, and MATRIX+RadISH [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Both JetVHeto and RadISH implement the same analytic resummation formula of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our research extends and improves upon these earlier results through detailed phenomenological studies of specific final states, including Higgs boson production [5, 12–14], W +W − production [15, 16], and ZZ and W ±Z production [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Another important aspect of our study is the performance of the resummation at N3LLp accuracy, which has not always been the case in 1The last missing ingredient for N3LL resummation is the exact dveto 3 (the three-loop rapidity anomalous dimension) which we approximate and take into account with an uncertainty estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We discuss this in detail in the subsequent section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 1 – previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We also describe our approximation of the missing dveto 3 that would be necessary to reach full N3LL accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Finally, we include our results in MCFM, a publicly distributed code, which allows users to easily perform studies in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Resummation of jet-veto logarithms has a close relationship with the resummation of transverse momentum logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the latter, one is interested in transverse momenta all the way down to zero pT , so the logarithms can be larger than in jet-veto processes where pveto T in the range 25 to 30 GeV is used experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In this paper we explore which jet-veto processes actually require resummation at these values of pveto T , supply the best predictions for those processes where it is warranted, and confront our theoretical predictions with experimental data where it is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In Section 2 we discuss the jet-veto factorization theorem including its ingredients that result in the resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We describe our setup for phenomenology including our uncertainty procedure in Section 3, compare with the public code JetVHeto in Section 4, and study the phenomenological implications for a wide range of processes in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 2 Jet-veto factorization and resummation We consider processes where jets have been defined using sequential recombination jet algorithms [18] with distance measure dij = min(k2n Ti, k2n Tj) ∆y2 ij + ∆φ2 ij R2 , diB = k2n Ti , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) where the choice n = −1 is the anti-kT algorithm [19], n = 0 is the Cambridge-Aachen algorithm [20, 21], and n = 1 is the kT algorithm [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' kTi denotes the transverse momentum of (pseudo-)particle i with respect to the beam direction, and ∆yij and ∆φij are the rapidity and azimuthal angle differences of (pseudo-)particles i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To describe the resummation method we focus on the simplest case of quark-antiquark induced Drell-Yan production of a lepton pair of invariant mass Q and rapidity y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The case of gluon initiated processes is structurally the same, but with different ingredients that we give below and in the appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the presence of a jet veto over all rapidities we have a factorization formula [3, 12, 13], d2σ(pveto T ) dQ2dy = dσ0 dQ2 ��CV (−Q2, µ) ��2 × � Bq(ξ1, Q, pveto T , R, µ, ν) B¯q(ξ2, Q, pveto T , R, µ, ν) S(pveto T , R, µ, ν) � + O �pveto T Q � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) where ξ1,2 = (Q/√s) e±y and, dσ0 dQ2 = 4πα2 3NcQ2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) – 2 – Table 1: Counting of orders in the resummation, adapted from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The second column indicates the nominal order when counting L⊥ ∼ 1/αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The third column states which logarithms are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The last three columns show the necessary additional anomalous dimensions and hard function corrections in each successive order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The requisite anomalous dimensions are provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Approximation Nominal order Accuracy ∼ αn s Lk ⊥ Γcusp γcoll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' H LL α−1 s 2n ≥ k ≥ n + 1 Γ0 tree tree NLL+LO α0 s 2n ≥ k ≥ n Γ1, γ0 tree N2LL+NLO α1 s 2n ≥ k ≥ max(n − 1, 0) Γ2 γ1 1-loop N3LL +NNLO α2 s 2n ≥ k ≥ max(n − 2, 0) Γ3 γ2 2-loop In this equation CV is a matching coefficient whose square is the hard coefficient function that corrects the lowest order cross-section, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Bq and B¯q are the quark beam functions which describe the emission of radiation collinear to the two beam directions in the presence of a jet veto, and S describes the emission of soft radiation in the presence of a jet veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The quantity ν is a supplementary scale necessitated by the rapidity divergences present in beam and soft functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The main process-independent ingredients are the beam and soft functions for both incoming quarks and gluons which have been published recently at the two-loop level [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The hard function is process specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We have used the existing two-loop fixed order implementations in MCFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Overall the factorization theorem achieves a separation of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The hard function contains logarithms of the ratio Q2/µ2, which can be minimized by setting µ2 = µ2 h ∼ Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, inside the beam and soft functions, it is natural to choose µ = pveto T to avoid large logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The resummation of large logarithms is achieved by choosing µ ∼ Q in the hard function and evolving it down to the resummation scale µ ∼ pveto T using the renormalization group (RG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For the hard function the evolution is solved analytically, see Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In RG-improved power counting the logarithms L⊥ = 2 log(µh/pveto T ), where µh is of order Q, are assumed to be of order 1/αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' With this definition the counting of powers of αs and of the large logarithm L⊥ is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The non-logarithmic terms that the resummation does not provide are easily accounted for by adding the matching corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The matching corrections are a finite contribution and add the effect of fixed-order corrections while removing the logarithmic overlap through a fixed-order expansion of the resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Soft function The jet veto soft function has been calculated using an exponential regulator [27] in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The calculation is divided into the sum of the soft function for a reference observable and a correction factor, S(pveto T , R, µ, ν) = S⊥(pveto T , µ, ν) + ∆S(pveto T , R, µ, ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) – 3 – In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [25] the reference observable is the transverse momentum of the color singlet system denoted by S⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' S⊥ can be derived from the expression given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [28, 29] after performing the Fourier transform to momentum space (see, for instance, the rules given in Table 1 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' ∆S depends on the jet algorithm and contributes for two or more emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' It thus depends only on double real emission diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 Refactorization and reduced beam functions For consistency with the transverse momentum resummation framework in CuTe-MCFM [31] we cast the factorization theorem in terms of the collinear anomaly framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In this framework the rapidity logarithms are exponentiated directly instead of resummed by solving rapidity RG equations [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For this we rewrite the square bracket in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) as follows, Bq(ξ1, Q, pveto T , R, µ, ν) B¯q(ξ2, Q, pveto T , R, µ, ν)S(pveto T , R, µ, ν) = � Q pveto T �−2Fqq(pveto T ,R,µ) e2hF (pveto T ,µ) ¯Bq(ξ1, pveto T , R, µ) ¯B¯q(ξ2, pveto T , R, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) The ν dependence vanishes in this combination of beam and soft functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We have factored out ehF/A(pveto T ,µ) from each beam function, resulting in the reduced beam functions ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' By construction hF/A are solutions of the RGE equation, d d ln µ hF/A(pveto T , µ) = 2ΓF/A cusp(µ) ln µ pveto T − 2γq/g(µ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6) with boundary condition hF/A(µ, µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The superscripts F or A signify whether the color is treated in the fundamental (F) or adjoint (A) representation, corresponding to a quark initiated process or a gluon initiated process, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The exact form of hF/A(pveto T , µ), determined by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6), is given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In terms of the reduced beam functions the jet-vetoed cross-section is now given by, d2σ(pveto T ) dQ2dy = dσ0 dQ2 ¯H(Q, µ, pveto T ) ¯Bq(ξ1, pveto T , R, µ) ¯B¯q(ξ2, pveto T , R, µ) + O(pveto T /Q) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7) The choice of hF/A in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6) divides Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) into two separately RG invariant pieces, the product of the two reduced beam functions ( ¯Bq ¯B¯q), and the hard function, ( ¯H) ¯H(Q, µ, pveto T ) = ��CV (−Q2, µ) ��2 � Q pveto T �−2Fqq(pveto T ,R,µ) e2hF (pveto T ,µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8) For quark-initiated processes the functions CV and Fqq obey the following RG equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' d d ln µ CV (−Q2, µ) = � ΓF cusp(µ) ln −Q2 µ2 + 2γq(µ) � CV (−Q2, µ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9) d d ln µFqq(pveto T , R, µ) = 2ΓF cusp(µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10) – 4 – Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10) are structurally the same for the gluon case with different anomalous dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The function ¯H is RG invariant due to the RGE’s satisfied by CV and Fqq and hF : d dµ ¯H(Q, µ, pveto T ) = O(α3 s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Consequently, the remaining product of reduced beam functions is also RG invariant, up to the order calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In our case, d dµ ¯Bq(ξ1, pveto T , R, µ) ¯B¯q(ξ2, pveto T , R, µ) = O(α3 s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11) The confirmation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11) and the confirmation of the R-dependence of the collinear anomaly given in the next section are two simple checks of the results of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Full details of the formulas needed to perform this check are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' If the scale pveto T is in the perturbative domain, the reduced beam function can be written in terms of the matching kernels ¯I as ¯Bi(ξ, pveto T , R, µ) = � j=g,q,¯q � 1 ξ dz z ¯Iij(z, pveto T , R, µ) φj/P (ξ/z, µ) , where φ denotes the usual collinear parton distribution of a parton of flavor j in a proton P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The matching coefficients ¯I are extracted from I, the two-loop beam and soft functions of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [24, 25] as, ¯Iij(z, pveto T , R, µ) = e−hF/A(pveto T ,µ) Iij(z, pveto T , R, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12) The coefficients in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [24] are presented as a Laurent expansion in the jet radius parameter R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Analytic expressions are presented for all flavor channels except for a set of R-independent non-logarithmic terms which are presented as numerical grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For our purposes we have interpolated the numerical grids using a spline fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We give further details on the reduced beam functions in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 The collinear anomaly coefficient and its approximations The missing ingredient for a complete N3LL resummation is the three-loop collinear anomaly coefficient and therefore warrants a longer discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This limitation has been discussed in the literature and approximated in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Here we discuss the uncertainty associated with the approximations and how we take it into account in our phenomenological predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' As presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10) the collinear anomaly coefficients obey the RG equations, d d ln µFqq(pveto T , R, µ) = 2ΓF cusp(µ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='13) d d ln µFgg(pveto T , R, µ) = 2ΓA cusp(µ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='14) – 5 – where, for example, Fqq has the expansion, Fqq(pveto T , R, µ) = αs 4πF (0) qq (pveto T , R, µ) + �αs 4π �2 F (1) qq (pveto T , R, µ) + �αs 4π �3 F (2) qq (pveto T , R, µ) + �αs 4π �4 F (3) qq (pveto T , R, µ) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15) While the logarithmic structure is given by the RG equations, the constant boundary parts dveto k (R, B) where B = F or A need to be determined by separate calculations and are also referred to as the rapidity anomalous dimensions in the framework of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 33]: F (0) qq (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µh) = ΓF 0 L⊥ + dveto 1 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F (1) qq (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µh) = 1 2ΓF 0 β0L2 ⊥ + ΓF 1 L⊥ + dveto 2 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F (2) qq (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µh) = 1 3ΓF 0 β2 0L3 ⊥ + 1 2(ΓF 0 β1 + 2ΓF 1 β0)L2 ⊥ + (ΓF 2 + 2β0dveto 2 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F))L⊥ + dveto 3 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F (3) qq (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µh) = 1 4β3 0ΓF 0 L4 ⊥ + (ΓF 1 β2 0 + 5 6ΓF 0 β0β1)L3 ⊥ + (1 2ΓF 0 β2 + ΓF 1 β1 + 3 2ΓF 2 β0 + 3dveto 2 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F)β2 0)L2 ⊥ + (ΓF 3 + 3dveto 3 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F)β0 + 2dveto 2 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F)β1)L⊥ + dveto 4 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='16) The analogous expression for gluons (F → A) is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The coefficients in the expansion of the cusp anomalous dimension, ΓF k , are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For single gluon emission dveto 1 (R, B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The function dveto 2 is defined below in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' There is only partial information on dveto 3 from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [14, 34, 35], and we have to rely on an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To estimate the validity of this approximation we first study similar approximations of dveto 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The function dveto 2 is given by [12], dveto 2 (R, B) = dB 2 − 32CB f(R, B) , where dB 2 = CB ��808 27 − 28ζ3 � CA − 224 27 TF nf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='17) The function f(R, B), which gives the dependence on the jet radius R, is known as an expansion about R = 0 up to terms including R4, f(R, B) = CB � − π2R2 12 + R4 16 � + CA � cA L ln R + cA 0 + cA 2 R2 + cA 4 R4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' � + TF nf � cf L ln R + cf 0 + cf 2R2 + cf 4R4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18) The terms on the first line are due to independent emission, whereas the terms on the second and third lines are due to correlated emission [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The expansion coefficients are given in Appendix D in analytic and numerical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 6 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Approximations for dveto 2 Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='17) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) we have for the gluon case in the limit nf → 0 and retaining only logarithmic and constant terms in R, dveto 2 (R, A) = −32C2 A � − 1 32C2 A dA 2 + cA L ln R + cA 0 � ≃ −32C2 A � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='096259 ln R + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7272641] ∼ 32C2 A × ln �R 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='19) This result was used as a basis for an approximation to dveto 3 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, the leading color (nf = 0) approximation is rather poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' With full nf dependence, but retaining only logarithmic and constant terms in R and setting nf = 5 we have dveto 2 (R, B) = 32CBCA � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='096 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0295nf) ln R − (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='72726 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12445nf) � ∼ 32CBCA � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2435 ln � R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='96 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='20) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 1 we show dveto 2 (R, A) and its approximations in units of dA 2 as a function of the jet radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' As a reminder, dA 2 is the non-R dependent part of d2, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We first compare the full result (red) with the inclusion of terms up order R2 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This shows that the R expansion converges quickly and it is sufficient to consider only terms up to R4 for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Including only the logarithm and the constant (blue) gives a reasonable approximation for sufficiently small R, with percent-level deviations around R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The leading color approximation (magenta) works only crudely as a first guess and could be used in the absence of any better estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 The function dveto 3 While the complete dveto 3 is unknown so far, we can extract the leading logarithmic term from results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Given that this approximation works reasonably well for dveto 2 for R ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4, it is reasonable to expect a similar behavior for dveto 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We further estimate the uncertainty associated with such an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15) the collinear anomaly coefficient at µ = pveto T is given by, Fgg(pveto T , R, pveto T ) = �αs 4π �2 dveto 2 (R, A) + �αs 4π �3 dveto 3 (R, A) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='21) Therefore, expanding the collinear anomaly we have that � Q pveto T �−2Fgg(pveto T ,pveto T ) =1 − 2 �αs(pveto T ) 4π �2 ln � Q pveto T � dveto 2 (R, A) − 2 ln �αs(pveto T ) 4π �3 ln � Q pveto T � dveto 3 (R, A) + O(α4 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='22) – 7 – Figure 1: Approximations of dveto 2 (R, A) scaled by the constant dA 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The full result, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='17) is plotted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The approxi- mation retaining only constant terms and logarithms of R is shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The ap- proximation retaining constant terms and logarithms of R and R2 terms is shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The leading color ansatz, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='19), derived setting nf = 0, is 32C2 A ln(R/2) and is shown in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The red, blue and green curves are all plotted for nf = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Figure 2: Effect of R0 variation in dveto 3 as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='24) with nf = 5, compared to the case dveto 3 = 0: R0 = 1 (black), R0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 (red, dashed), R0 = 2 (blue, dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' At order α3 s the leading term in the limit R → 0 can be extracted from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [14] which reads, Fcorrel LLR,31(R) = �αs 4π �3 ln � Q pveto T � 128CA ln2 R R0 × � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='803136C2 A − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='589237nf2TRCA + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='36982CF nf2TR − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='05893n2 f4T 2 R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='23) Comparing the third-order coefficient in the two equations we thus have for a general color representation dveto 3 (R, B) = −64CB ln2 � R R0 � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='803136C2 A + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='36982CF nf − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='589237CAnf − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='05893n2 f) = −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='38188 × 64CB ln2 � R R0 � for nf = 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='24) Hence, the sign of the leading term in the small R limit is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In this limit dveto 3 leads to an increase in the cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This approximation only gives the leading R behavior, and it has been suggested that one may plausibly take 1 2 < R0 < 2 as an uncertainty envelope [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Since dveto 3 enters through the collinear anomaly as an overall factor, we consider the impact of varying R0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For typical values of pveto T = 30 GeV (as considered in this paper for the – 8 – Table 2: Input and derived parameters used for our numerical estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' MW 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='385 GeV ΓW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0854 GeV MZ 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1876 GeV ΓZ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4952 GeV Gµ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='166390 × 10−5 GeV−2 mt 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 GeV mh 125 GeV m2 W = M2 W − iMW ΓW (6461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='748225 − 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='634879 i) GeV2 m2 Z = M2 Z − iMZΓZ (8315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='17839376 − 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='53129952 i) GeV2 cos2 θW = m2 W /m2 Z (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7770725897054007 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='001103218322282256 i) α = √ 2Gµ π M2 W (1 − M2 W M2 Z ) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='56246890198475 × 10−3 giving 1/α ≈ 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' comparison with experimental studies) there is an effect of less than two percent for R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This is in agreement with the deviations we found for dveto 2 for this approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We take into account this variation in our uncertainty estimates, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A definitive statement on this issue will have to await an exact calculation of dveto 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 3 Setup for phenomenology Before discussing phenomenological results, we list our input parameters, the method for matching to fixed order, and the approach for estimating uncertainties at fixed order and at the resummed level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Input parameters The input values used in our numerical studies are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' As indicated in the table we use the complex mass scheme for the W and Z boson masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The number of light quarks, nf, is set equal to five, except for the case of W +W −-production where nf = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We use the PDF distribution NNPDF31_nnlo_as_0118 except for W +W − where we use NNPDF31_nnlo_as_0118_nf_4 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Note that we use these NNLO parton distributions even in our lower order predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the cases of WW and ZZ production, at O(α2 s) the cross-section receives contributions from processes with two gluons in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' When performing the resummed calculations we only include such contributions at NLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, these contributions only represent about 3% of the cross-section for pveto T = 10 GeV, rising to about 6–8% for pveto T = 60 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Therefore, neglecting higher order corrections to these contributions, which are not implemented in our code, is justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Although only strictly true for the leading q¯q component we refer to the full resummed calculation as N3LLp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We match the resummation and fixed-order NkLO corrections using a naive additive scheme as – 9 – follows, σN(k+1)LL+N(k)LO(pveto T ) = σN(k+1)LL(pveto T ) + σ∆,k(pveto T ) , where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) σ∆(pveto T ) = σNkLO(pveto T ) − dσN(k+1)LL(pveto T ) ���� exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' to NkLO .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) The matching correction σ∆(pveto T ) is defined as a function of pveto T , using the difference between the fixed-order contribution and the resummed result expanded to the same fixed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The limit pveto T → 0 of σ∆(pveto T ) is finite, which also allows its use as a higher-order subtraction scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The use of a naive matching without a transition mechanism that switches off the resummation at large pveto T is justified since the matching corrections for all considered cases in this paper are small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' even in the most extreme case they are less than 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In other words, the resummation alone provides a good description of the cross-sections and does not need to be switched off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Any transition function to turn off the resummation at large pveto T would have a very small effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This is in contrast to transverse-momentum resummation where a transition function is necessary [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 Uncertainty estimates at fixed order Ultimately the resummed predictions should offer a practical advantage compared to the fixed-order predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In many cases, the quantity log(Q/pveto T ) is not very large, and it may not seem worthwhile to use resummed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, as we will show, the resummation works remarkably well on its own and has matching corrections of only up to around 20%, often much less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The clear separation of scales and the resummation then allow for smaller and more reliable uncertainty estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To set the stage, we first examine perturbative convergence and uncertainties at fixed order for quark and gluon induced boson processes, as well as for WW and ZZ production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Constructing jet-vetoed cross-sections at fixed order requires the combination of different cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, if we naively subtract the jet cross-section from the inclusive result, it can result in underestimated uncertainties and narrowing uncertainty bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To avoid this, different methods have been proposed in the literature, of which we compare the following two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' One strategy, which we term the "two-scale" approach, is to consider the different relevant scales Q and pveto T of the vetoed cross-section σ0, and include both of them in the uncertainty estimate through a multi-point variation around both scales [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To compute this uncertainty, we separately vary the renormalization scale µr and the factorization scale µf over the values {µh, 2µh, µh/2, pveto T , 2pveto T , pveto T /2}, where µh depends on the process under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' An estimate of the uncertainty is then obtained by adding in quadrature the maximum deviations from µr = µf = µh, from µr and µf variation separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 10 – Another approach, advocated by Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [14, 37], takes the jet-veto efficiency (JVE) as the central quantity, which is the ratio of jet-vetoed cross-section to total cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' By combining the uncertainties of these two quantities in quadrature, one obtains a more robust estimate of the uncertainty in the jet-vetoed cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This is because the uncertainties are considered uncorrelated: the uncertainties in the jet-veto efficiency are typically due to non-cancellation of real and virtual contributions, while those in the total cross-section are connected with large corrections from higher orders [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For our JVE approach, we follow the simplest formulation (“scheme (a)” of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [14]) to compute a JVE-based uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For this we consider variation over the scales {µh, 2µh, µh/2} of σincl and combine in quadrature the uncertainty from the calculation of the 0-jet efficiency (σ0/σincl) and the uncertainty from the inclusive calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our final fixed-order uncertainty band is the envelope of the two-scale and JVE approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' With these procedures, our fixed-order results for Z and H production are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For Z production we use the canonical choice µh = Q, where Q is the invariant mass in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For Higgs production we use µh = Q/2, guided by the calculation of the inclusive cross-section where such a choice results in markedly-improved perturbative convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We observe that for Z production the NNLO uncertainty band is wholly contained within the NLO one, while for the Higgs case the bands at least overlap somewhat throughout the range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For Higgs production following the combined two-scale and JVE approach results in a significantly larger uncertainty at both NLO and NNLO, especially at smaller values of pveto T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' On the other hand, for Z production the additional uncertainty from the JVE approach is very small and negligible at NNLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Predictions for WW and ZZ production (with µh = Q) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The limited overlap between the NLO and NNLO bands indicates that uncertainties are underestimated, even with the generous scale uncertainty procedure that we follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The additional uncertainty resulting from the JVE procedure is small, especially at NNLO, because the scale uncertainty of the inclusive cross-sections is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 Uncertainty estimates at the resummed and matched level For our central predictions, we set the resummation and factorization scales to µ = pveto T and the hard scale (corresponding to the renormalization scale) to µh = Q, where Q is the invariant mass of the color-singlet final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The exception is Higgs production, where we choose µh = Q/2 as previously discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For the collinear anomaly coefficient dveto 3 , we use the form given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='24) [14] with R0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Complications arising at fixed order, described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2, are not present in the resummed case and therefore we can follow a simpler approach where we vary all scales in our formalism and take the envelope, as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' While the matching of resummed predictions to fixed-order could still introduce a complication, the matching corrections are not dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The bulk of the cross-section comes from the resummation and it allows us to follow the simple – 11 – (a) Z production using the setup of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (b) H production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Figure 3: Comparison of NLO and NNLO fixed order predictions as a function of the jet veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Central predictions solid, uncertainty estimates using either the two-scale approach (dotted) or the envelope of that and the JVE approach (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (a) WW production using the setup of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (b) ZZ production using the setup of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Figure 4: Comparison of NLO and NNLO fixed order predictions as a function of the jet veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Central predictions solid, uncertainty estimates using either the two-scale approach (dotted) or the envelope of that and the JVE approach (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 12 – procedure of varying all scales in the naively obtained (without JVE) jet-veto cross-section too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The small and narrowing uncertainty bands at fixed order would typically appear in regions where the resummation is found to be dominant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' where fixed-order contributes very little through the matching corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In practice we observe that the size of uncertainties are overall uniform in both the resummation and large pveto T fixed-order regions, as can be seen in all of our following predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This supports the conclusion that our procedure is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Overall, our procedure for estimating uncertainties is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For the resummation (fixed-order) parts we vary both the resummation (factorization) and hard (renormalization) scales by a factor of two about their central values, adding the excursions in quadrature to obtain the total scale uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For the resummation we re-introduce the rapidity scale in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) by re-writing the collinear anomaly factor as follows [12, 41]: � Q pveto T �−2Fii(pveto T ,R,µ) = �Q ν �−2Fii(pveto T ,R,µ)� ν pveto T �−2Fii(pveto T ,R,µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) For ν ∼ pveto T the second factor can be expanded since it does not contain a large logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We vary the rapidity scale ν in the range [pveto T /2, 2pveto T ] for gluon-initiated processes and in the range [pveto T /6, 6pveto T ] for quark-initiated processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The large variation for quark-initiated processes ensures overlapping uncertainty bands at NNLL and N3LLp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' this is achieved by the range given above, as demonstrated explicitly in Sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The parameter R0 in dveto 3 is varied between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We first combine the scale uncertainties (1 and 2) in quadrature and then, to obtain our total uncertainty, add the variation of R0 (3) linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 Effects of cuts on rapidity at fixed order The usual jet veto resummation described so far imposes no cut on the jet rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This is in contrast to experimental analyses, see Table 3, which impose such a cut because of limited detector acceptance and to diminish the effect of pileup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [42] identifies three different regimes, depending on pt, Q and ycut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For pveto T /Q ≫ exp(−ycut) standard jet veto resummation should apply, effects due to the rapidity cut are corrections power suppressed by Q exp(−ycut)/pveto T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For pveto T /Q ∼ exp(−ycut) the effects of a rapidity cut must be treated as a leading power correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For pveto T /Q ≪ exp(−ycut) the logarithmic structure is changed already at leading log level, and non-global logarithms appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 13 – Table 3: Jet rapidity cuts applied in the experimental studies examined later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Process Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' ycut Higgs – no study Z (CMS) [38] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 W (ATLAS) [43] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 WW (CMS) [39] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 WZ (ATLAS) [44] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 WZ (CMS) [45] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 ZZ (CMS) – no study (a) Z production following the setup of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (b) H production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Figure 5: Effect of the jet rapidity cut at NNLO with pveto T = 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We estimate the practical impact of experimentally used jet rapidity cuts at fixed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Including the rapidity cut in the resummation requires large changes and ingredients, which are also only available a low order so far [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The effect of the jet rapidity cut for the Z and Higgs production cases is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' These calculations are performed at NNLO for pveto T = 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The rapidity cut plays a bigger role for Higgs production: for example for ycut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 the cross-section is 11% larger than the result with no rapidity cut, compared to only 2% for Z production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This is due to the larger logarithm (log(mH/pveto T )/ log(mZ/pveto T ) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='28) and the larger color prefactor (CA/CF = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='25) in Higgs production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, for ycut = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 the effect of the rapidity cut is negligible in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 14 – (a) WW production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (b) ZZ production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Figure 6: Effect of the jet rapidity cut at NNLO with pveto T = 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The corresponding results for diboson processes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In this case, the disparity between Q and pveto T is much larger, so the rapidity cut can play a crucial role, although the effect is still not as important as for Higgs production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For ycut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 the WW and ZZ cross-sections 4% larger than the results with no rapidity cut, and the effect of ycut = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 4 Comparison with JetVHeto While jet-veto resummed phenomenology has been extensively studied in the literature, the only public codes that permit detailed predictions use JetVHeto or RadISH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For jet-veto resummation RadISH implements the analytic JetVHeto resummation formula [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The codes rely on the formalism of the CAESAR approach [4, 46] extended to NNLL [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' An extension of the RadISH code has been used to perform joint jet-veto and boson transverse momentum resummation [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For our comparisons we use RadISH version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0 [48, 49] and JetVHeto version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0 [5, 14, 37] including small-R resummation [4, 35] as part of MCFM-RE [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Both codes operate at the level of NNLL and we have checked that they give indeed the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In our comparison, we would like focus on the differences in the resummation part, since the fixed-order part is identical in each calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We explore how central values and uncertainties compare at NNLL to our results and in how far N3LLp results improve the perturbative convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, the matching to fixed-order is handled differently in each formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Different matching schemes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' additive or multiplicative schemes of various – 15 – types) probe higher-order effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' It has also been advocated to match at the level of jet-veto efficiencies [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Fortunately, matching corrections are generally small for jet-veto scales of 30 to 40 GeV for all considered boson and di-boson processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We therefore focus on the resummation in our comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The JetVHeto formalism considers three scales µR, µF and Q that are all similar in magnitude to the hard scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To ensure that the resummation switches off for pveto T ≳ Q, the resummed logarithms are modified through the prescription log(Q/pveto T ) → 1/p log((Q/pveto T )p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For JetVHeto p has a default value of 5 [14], while for RadISH the default choice is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For comparison purposes we use p = 5 in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' It is evident that for sufficiently small pveto T the precise value of p does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Changing this parameter has a similar effect to turning off the resummation with a transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In principle this demands a fully matched calculation, but the matching corrections of our considered cases are small and we have checked that the effect of changing p to 3 or 4 is subleading compared to the scale uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Here we focus on those scale uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [14] it has been argued that the Q should be varied by a factor of 3 2 around its central value, based on new insights from convergence at N3LO for Higgs production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For simplicity, we use a more conservative variation by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We independently vary µR, µF and Q by a factor of two around a central scale of mℓℓ for Z-boson production and around mH/2 for Higgs production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our uncertainty bands for this comparison are obtained by taking the envelope of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Z-boson production For the comparison of Z production we choose a central hard scale of mℓℓ with results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We find that our MCFM NNLL central values have only marginal compatibility with our JetVHeto uncertainty estimates, despite having the same logarithmic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This indicates that the JetVHeto uncertainties (as estimated according to our procedure just described) do not fully account for the higher-order corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' On the other hand, our uncertainties at NNLL are larger, leading to an overall agreement between the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' At N3LLp uncertainties decrease dramatically compared to NNLL, but they are quite asymmetric, which suggests that a symmetrization of uncertainties may be necessary in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We also observe that without the large uncertainties at NNLL, there would be no overlap between the N3LLp results and NNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This highlights the importance of carefully estimating and comparing uncertainties to accurately assess the compatibility of different methods and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' H-boson production In our study of Higgs production, we choose a central hard scale of mH/2 and show results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' All results are computed in the mt → ∞ theory and rescaled by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0653 to account for finite top-quark mass effects, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 16 – qq → Z → e+e−, s = 13 TeV, µh = me+e− 200 400 600 800 10 20 30 40 pt veto [GeV] σveto [pb] RadISH/JetVHeto/MCFM−RE NNLL MCFM NNLL MCFM N3LLp Figure 7: Comparison of JetVHeto NNLL resummation with our NNLL and N3LLp results for Z production with cuts as in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The Higgs case is distinct from Z production since it is gluon-gluon initiated instead of quark-initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In this case, our predictions agree well with the JetVHeto results, but our uncertainties at NNLL are again much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Note that we vary the JetVHeto scale Q by a factor of two, while the JetVHeto authors vary by a factor of 3/2 in the Higgs case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This difference in the amount of variation may require some tuning in our formalism, at least at the NNLL level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, the perturbative convergence is again excellent with small uncertainties at N3LLp and central predictions that agree well with NNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5 Phenomenological results In this section, we present the results of our phenomenological studies, which are based on the uncertainty procedure, matching to fixed-order, and input parameters described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We compare our findings with experimental results from the literature and discuss their implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Z and W production The process of Z production has already been extensively studied in the literature, thus enabling a variety of cross-checks of our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The implementation of the hard function and its evolution has been verified by comparison with the explicit results given in Table 1 of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The full machinery of the resummation and matching procedure can also be compared with the results of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [5], with which we find excellent agreement within uncertainties, see also Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 17 – gg → H, s = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 TeV, µh = mH 2 20 30 40 20 30 40 50 pt veto [GeV] σveto [pb] RadISH/JetVHeto/MCFM−RE NNLL MCFM NNLL MCFM N3LLp Figure 8: Comparison of JetVHeto NNLL resummation with our NNLL and N3LLp results for non-decaying H production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Table 4: Cuts used in the analysis of Z production, adapted from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' lepton cuts ql1 T > 30 GeV, ql2 T > 20 GeV, |ηl| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 lepton pair mass 71 GeV < ml−l+ < 111 GeV jet veto anti-kT , R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4, 0-jet events only We first investigate the impact of choosing a time-like hard scale in the resummed result for Z production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Previous work has shown that choosing a space-like hard scale (µ2 h = Q2) can lead to significant corrections in the perturbative expansion of some processes, while a time-like hard scale (µ2 h = −Q2) can resum certain π2 contributions [51] using a complex strong coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For this comparison we consider purely resummed results at NNLL and N3LLp, only considering uncertainties originating from scale variation (items 1 and 2 of our uncertainty procedure in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We consider the process pp → Z/γ∗ → ℓ−ℓ+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' a final state of definite lepton flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We use the same set of cuts and vetoes as in the √s = 13 TeV CMS analysis [38], but extend the veto to jets of all rapidities, rather than only those with |y| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This difference, and the effect of matching to NNLO, is discussed in detail in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 9a as a function of the value of the jet veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We observe that the results do not depend strongly on the choice of hard scale, with a difference of about 4% at NNLL and only 1% at N3LLp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This indicates that resumming the π2 terms results in only – 18 – (a) Predictions are computed using a central choice for the hard scale given by either µ2 h = Q2 or µ2 h = −Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The lower panel shows the ratio of the result for µ2 h = −Q2 to the one for µ2 h = Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (b) Predictions and CMS measurement as ratio to matched result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Figure 9: Comparison of NNLL and N3LLp predictions for Z production as a function of the jet veto, using the setup of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [38] (central predictions solid, uncertainty estimate according to the text, dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' a small enhancement of the cross-section for W and Z production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Based on these findings, we use the space-like hard scale (µ2 h = Q2) in our subsequent studies of Z and W boson production, as it is the more commonly used choice in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 CMS Z production As previously mentioned, the CMS measurement we are comparing to includes a jet rapidity cut of |y| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To assess the importance of this restriction, we first compare the NNLO predictions with and without the rapidity cut, as a function of the jet veto value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This comparison, shown in Table 5, helps us better understand the limitations of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We use the quantity ϵ(pveto T ) to quantify the increase in the cross-section when the rapidity cut is applied, defined as ϵ(pveto T ) = σ0−jet(ycut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) σ0−jet(no ycut) − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) The experimental measurement we are comparing to uses a jet veto of pveto T = 30 GeV, for which the rapidity cut has only a 3% effect on the cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This suggests that our calculation with an all-rapidity jet veto is appropriate for comparing to the experimental measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, as pveto T decreases, the impact of the rapidity cut becomes more significant, until at – 19 – Table 5: The Z + 0-jet cross-section prediction at NNLO (µ = Q), with and without a jet rapidity cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' pveto T [GeV] 5 10 20 30 40 σ0−jet(no ycut) [pb] 140 347 539 627 675 σ0−jet(ycut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) [pb] 242 411 569 643 685 ϵ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='01 qq → Z → l+l−, s = 13 TeV, CMS cuts, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='02872 618 ± 17 pb 592−13 +9 pb 600 650 700 750 800 CMS NLO NNLL NNLL+NLO NNLO N3LLp N3LL+NNLO σveto [pb] Figure 10: Comparison of Z-boson jet-vetoed predictions with the CMS [38] 13 TeV measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Shown are results at fixed-order, purely resummed and matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' pveto T = 5 GeV it is no longer appropriate to neglect the rapidity cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This is consistent with the arguments of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [42], which suggest that the standard jet veto resummation formalism should suffice as long as ln(Q/pveto T ) ≪ ycut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In our case, ln(Q/pveto T ) ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 for pveto T from 40 down to 5 GeV, so the standard jet veto resummation should be appropriate, albeit with sizeable power corrections, for ycut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 except for the smallest values of pveto T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We now turn to a comparison with the CMS result [38], which uses a jet threshold of 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our comparison with fixed-order, purely resummed and matched predictions is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We find that the fixed-order and resummed results differ by only a few percent, indicating that resummation is not necessary for this value of the jet veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This is because the quantity ln(MZ/pveto T ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 is not large enough to require resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The CMS measurement yields a cross-section of 618 ± 17 pb, while our best prediction is 592+9 −13 pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We study the production of Z bosons as a function of the jet veto in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We observe that the difference between the resummed and central fixed-order results is small, even for the smallest values of pveto T considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, the uncertainties in the fixed-order prediction are larger across the whole range, particularly for small pveto T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For values of pveto T in the range of 20 to 40 GeV, which are of practical interest, the N3LLp uncertainty is smaller than the NNLO uncertainty by about a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 ATLAS W production We now perform a comparison with √s = 8 TeV ATLAS data on W production [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For this study, jets were identified using the anti-kT algorithm with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content="4 and must satisfy – 20 – qq' → W± → eν, s = 8 TeV, ATLAS cuts, arXiv:1711." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='03296 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 nb 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='71−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='07 nb 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0 ATLAS NLO NNLL NNLL+NLO NNLO N3LLp N3LL+NNLO σveto [nb] Figure 11: Comparison of W-boson jet-vetoed predictions with the ATLAS [43] 8 TeV mea- surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Shown are results at fixed-order, purely resummed and matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' pT > 30 GeV and |y| < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We have checked at fixed order that this large rapidity cut has a negligible impact of a few per mille, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' results are unchanged within the numerical precision to which we work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Summing over both W charges and including only the decay into electrons we compare our predictions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We show results at fixed order, at the resummed level, and at the matched level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The effect of matching is large and we thus conclude that this value for the jet veto is outside the sensible range for a purely resummed result, unlike for the Z study in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We observe excellent agreement with the theoretical prediction, albeit with a larger experimental uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The experimentally measured cross-section is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='30 nb while our best prediction is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='71+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10 nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Since this measurement corresponds to an integrated luminosity of only 20 fb−1 it is clear that the high-luminosity LHC will eventually be able to provide a much keener test of perturbative QCD in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 W +W − production Experimental studies of WW production were performed by both ATLAS [52, 53] and CMS [39, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Here we focus on the CMS analysis of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [39] since it provides a measurement of the 0-jet cross-section as a function of the jet pT veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This cross-section measurment corresponds to a sum over both electron and muon decays of the W bosons, which we denote by the label pp → W −W + → 2ℓ2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In order to account for this in our calculation, we compute the result for pp → e−µ+¯νeνµ at NNLO and multiply it by the factor that accounts exactly for all lepton combinations through NLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The impact of ZZ contributions in the same-flavor case results in a slight enhancement over the naïve factor of four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We find that, independent of the value of the jet veto in the range that we consider, this factor is equal to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The CMS analysis only imposes a jet rapidity cut of ycut = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5, so our expectation is that the standard jet veto resummation formalism should be appropriate for pveto T values between 60 and 10 GeV, since in this case the logarithm of the ratio of Q to pveto T are in the range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This expectation is supported by the NNLO analysis in Table 6, which shows only a – 21 – small 2% effect from the rapidity cut for pveto T = 10 GeV (and none for values above that).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Unlike the processes considered so far, Q is no longer set by a resonance mass but is instead a distribution with a peak slightly above the 2MW threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For illustration, we have used an average value of Q ∼ 220 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We first fix the value of pveto T = 30 GeV and study the sensitivity of the pure fixed-order and resummed calculations to the jet-clustering parameter R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' At NLO, there is at most one additional parton, so the NLO result does not depend on the value of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, the NNLL result exhibits a mild dependence on R, which is most noticeable in the size of the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' These uncertainties are much larger for smaller values of R, as was previously observed and discussed in the context of Higgs production in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' At NNLO, the fixed-order calculation becomes sensitive to the value of R, although the dependence is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' At N3LLp, the dependence is reduced compared to NNLL, especially at small R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Overall, these results suggest that the jet-clustering parameter has a mild effect on the predictions of the fixed-order and resummed calculations for WW production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We have not investigated the effect of small R resummation [14] on these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 12b, we extend our previous analysis of the jet-veto dependence of WW production, which was presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The effect of matching is substantial for values of pveto T greater than 20 GeV, so for typical jet vetoes in the range of 20 to 40 GeV, matched predictions are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We find that the fixed-order description is only capable of providing an adequate result for the highest value of pveto T studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A comparison with the CMS measurement shows better agreement with the matched resummed calculation, although the experimental uncertainties are still substantial, corresponding to an integrated luminosity of 36 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We eagerly anticipate a measurement with more statistics in order to hone this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Future measurements with higher precision and larger data samples will provide a more stringent test of the theoretical predictions and help to refine our understanding of WW production at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 W ±Z production 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 ATLAS For W ±Z production, we first compare our results with an analysis from the ATLAS collabora- tion at √s = 13 TeV [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The 0-jet cross-section is measured with jets defined by the anti-kT Table 6: The pp → W −W + → 2ℓ2ν+0-jet cross-section at NNLO, with and without a jet rapidity cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' pveto T [GeV] 10 25 30 35 45 60 σ0−jet(no ycut) [fb] 535 963 1004 1054 1145 1237 σ0−jet(ycut = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) [fb] 548 963 1004 1054 1145 1237 ϵ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00 – 22 – (a) Jet radius R dependence of fixed-order and purely resummed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (b) Predictions and CMS measurement as a ratio to the matched result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Figure 12: Comparison of NNLO, N3LLp and matched N3LLp+NNLO results for W +W − production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=" qq' → W±Z, s = 13 TeV, ATLAS cuts, arXiv:1902." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='05759 31 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 fb 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 fb 27 30 33 36 ATLAS NLO NNLL NNLL+NLO NNLO N3LLp N3LL+NNLO σveto [pb] Figure 13: Comparison of W ±Z jet-vetoed predictions with the ATLAS 13 TeV measurement [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Shown are results at fixed order, purely resummed and matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' algorithm with pT > 25 GeV, |y| < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5, and R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Since ln(Q/pveto T ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 (for pveto T = 25 GeV, using an average Q of about 240 GeV), we expect that standard jet veto resummation should be applicable in this case, since ycut = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We have checked that the effect of the rapidity cut is at the per mille level, which is less than our numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The ATLAS result is presented for a single leptonic channel and summed over both W charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The corresponding theoretical predictions at fixed order, at the resummed level, and at the matched level are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=" – 23 – qq' → W±Z, s = 13 TeV, CMS cuts, arXiv:2110." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11231 166 ± 6 fb 128 ± 8 fb, ycut < ∞ 120 140 160 180 CMS NLO NNLL NNLL+NLO NNLO N3LLp N3LL+NNLO σveto [pb] Figure 14: Comparison of W ±Z jet-vetoed predictions with the CMS [45] 13 TeV measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Shown are results at fixed-order, purely resummed and matched, all without a rapidity cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Overall, the measurement is in good agreement with both the N3LLp+NNLO and NNLO predictions, within the mutual uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Only a more precise measurement would be able to definitively support the need for resummation in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Since the ATLAS analysis includes only 36 fb−1 of data, it is likely that a more precise measurement will be possible in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 CMS We now contrast the ATLAS study of the W ±Z process with one from CMS [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the CMS study, jets are defined by the anti-kT algorithm with pT > 25 GeV, |y| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5, and R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To assess the applicability of the jet-rapidity inclusive resummation framework, we must com- pare ln(Q/pveto T ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 with ycut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This suggests that the standard jet veto resummation formalism may not be appropriate in this case, and that the use of ycut-dependent beam functions [42] may be necessary to provide a reliable theoretical prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Despite this, we still pursue the comparison here, without using ycut-dependent beam functions, to examine the limitations of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The CMS result for W ±Z production is presented after summing over all lepton flavors and both W charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' On the theoretical side, we perform a similar analysis, but ignore same-flavor effects that only enter at the 2% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To construct the jet-vetoed cross-section for the CMS measurement, we combine the differential results in Figure 14(c) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [45] with the inclusive cross-sections reported in Table 6 of the same reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We find that neither the resummed prediction nor the NNLO one are in good agreement with the CMS data, even when the NNLO calculation takes the jet rapidity cut into account (increasing the NNLO result from 128 fb to 137 fb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This suggests that resummation is required in this case, and that the use of ycut-dependent beam functions is necessary to provide a reliable theoretical prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Overall, these results highlight the importance of using appropriate resummation techniques to accurately predict W ±Z production at the LHC with a small jet rapidity cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 24 – lepton cuts ql1 T > 20 GeV, ql2 T > 10 GeV, ql3,4 T > 5 GeV, |ηl| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 lepton pair mass 60 GeV < ml−l+ < 120 GeV jet veto anti-kT , R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 Table 7: Fiducial cuts used for the ZZ analysis, taken from the CMS study in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Table 8: The ZZ + 0-jet cross-section at NNLO (µ = Q), with and without a jet rapidity cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' pveto T [GeV] 10 20 30 40 50 60 σ0−jet(no ycut) [fb] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 σ0−jet(ycut = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) [fb] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 σ0−jet(ycut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) [fb] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8 ϵ(ycut = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00 ϵ(ycut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 ZZ production In the absence of jet-vetoed cross-sections for comparison, we use the cuts from a recent CMS study [40] to investigate our theoretical predictions for ZZ production as a function of pveto T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the results that follow we consider a sum over Z decays into both electrons and muons, which we denote by pp → ZZ → 4 leptons, and apply the cuts shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We expect that standard jet veto resummation should provide good predictions for ycut = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5, since ln(Q/pveto T ) is in the range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 for pveto T values between 60 and 10 GeV, using an average Q of about 240 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For ycut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5, we expect larger rapidity effects for the smallest values of pveto T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This is supported by our analysis in Table 8, which shows only a very small (1%) effect from a rapidity cut of ycut = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 for pveto T = 10 GeV (and no effect for higher values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Even for ycut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5, the rapidity cut has a relevant effect only for pveto T values below 30 GeV, and is mostly insignificant beyond that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 15a shows a comparison of the dependence on pveto T for purely-resummed results at two different logarithmic orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The central predictions are very similar at NNLL and N3LLp and are consistent within uncertainties for all values of pveto T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 15b compares the matched N3LLp+NNLO and NNLO results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The NNLO prediction has large uncertainties over the whole range of pveto T and only overlaps with N3LLp+NNLO around 40 GeV and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The difference between the central resummed and fixed-order results is significant (around 10%) for typical values of pveto T around 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For most relevant values of pveto T at the LHC, resummation is clearly important for providing a precision prediction for this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5 Higgs production For gluon fusion Higgs production an important topic is the inclusion of finite top-quark mass effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Although at NNLO these could be included exactly [56, 57], the mass effects are not relevant in the jet-vetoed case [58] at the current level of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A simple overall one-loop – 25 – (a) Purely resummed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (b) Ratio to matched result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Figure 15: Comparison of NNLO, N3LLp and matched N3LLp+NNLO results for ZZ production as a function of the jet veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' rescaling factor that takes into account the full mass dependence is sufficient to introduce mass effects into mt → ∞ EFT predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the resummation formalism, the coefficient for the matching of Higgs production in QCD onto SCET can be calculated in two ways, referred to as one-step and two-step procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 One-step and two-step schemes The one-step procedure is based on the observation that the ratio mH/mt is not large in a logarithmic sense (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' ρ = m2 H/m2 t ≈ 1/2 and αs log 1/ρ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='07).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This procedure matches the full QCD result, typically obtained at higher orders as an expansion in the parameter r, onto SCET at the scale µh ∼ mH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In this way, terms of order ρ are retained, but logs of mt/mH are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the two-step procedure outlined in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [59–62], the top quark is first integrated out at a scale µt ≊ mt, and then the QCD effective Lagrangian is matched onto the SCET at a scale µh ≊ mH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Running between µt and µh allows one to sum logarithms of mt/mH, and finite top-mass effects are included by scaling the result by a correction factor obtained at leading order (an increase with respect to the EFT result by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0653, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Terms enhanced by powers of mH/mt are thus only included in an approximate fashion at NLO and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The one-step procedure is described in detail in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 and the two-step procedure is described in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We compare the numerical difference between the one- and two-step schemes, computed at √s = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 TeV and for R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 16a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Guided by fixed-order results, and in accord with – 26 – (a) Results in the one- or two-step scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The lower panel shows the ratio of the one-step to the two-step result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (b) Results using a central scale of either µ2 h = Q2 or µ2 h = −Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The lower panel shows the ratio of the result for µ2 h = −Q2 to the one for µ2 h = Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Figure 16: Comparison of NNLL and N3LLp predictions for Higgs production at √s = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 TeV as a function of the jet veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' previous studies of this process [14], we set the hard (renormalization) scale using µh = Q/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We observe that the one-step scheme results in a cross-section that is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3% larger at NNLL and only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6% larger at N3LLp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This small difference occurs if one works rigorously at a fixed order of αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Working at a fixed order in αs in the component parts of the two-step scheme can lead to larger differences, as described in more detail in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 Time-like vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' space-like µ2 h We now study the impact of choosing a time-like hard scale for the calculation of the Higgs cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To do this, we compare µ2 h = (Q/2)2 (the space-like scale) with µ2 h = −(Q/2)2 (the time-like scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The use of a time-like hard scale allows us to resum certain π2 terms, by employing a complex strong coupling [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For this comparison, we consider purely resummed results at NNLL and N3LLp accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 16b, for the two-step scheme computed at √s = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 TeV with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We observe that at NNLL, the resummation of the π2 terms significantly enhances the cross-section by 17%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, at N3LLp accuracy, this resummation only leads to a small increase of 2% in the cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Results for the matched vetoed cross-section are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' After matching, we observe substantial agreement between the NNLO and N3LLp+NNLO calculations within – 27 – Figure 17: Comparison of NNLL, N3LLp and N3LLp+NNLO predictions for Higgs production at √s = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 TeV as a function of the jet veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The central predictions differ by about 5% across the range, but the uncertainties are substantially smaller in the resummed calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 6 Conclusions We have presented a comprehensive study of jet-veto resummation in the production of color singlet final states using the most up-to-date theoretical ingredients and achieving N3LLp accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our implementation in MCFM improves upon previous public NNLL calculations by reducing theoretical uncertainties, as demonstrated by comparisons with ATLAS and CMS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Once the one remaining theoretical element, dveto 3 , becomes available, it will be simple to upgrade our predictions to full N3LL accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 The primary motivation for this work comes from the need for reliable and accurate predictions of jet-veto cross-sections in processes such as Higgs boson and W +W − production, which are commonly used to study new physics at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In these processes, the imposition of a jet veto is often necessary to suppress backgrounds and enhance sensitivity to new physics signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Experimental results going beyond these two processes are much less frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We encourage the experimental collaborations to consider measurements of more Standard Model processes with a jet veto, as larger data samples become available, to better understand the dependence of these processes on the jet veto parameters pveto T and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 2We have shown that the effect of including gluon-induced process in W +W − and ZZ production is numerically a small effect, so that NLL accuracy is sufficient for these sub-processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 28 – In addition to providing improved predictions for jet-veto cross-sections, our work also serves as a valuable tool for testing and validation of general purpose shower Monte Carlo programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our code allows for a detailed investigation of the dependence on the jet parameters pveto T and R, providing a benchmark for assessing the logarithmic accuracy and reliability of Monte Carlo simulations in this important class of processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our analysis shows that at the currently experimentally used values of pveto T in W and Z production, the logarithms are not large enough to justify the use of jet-veto resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In these cases, fixed-order perturbation theory, which can be used to give the results with a jet veto over a limited range of rapidities, is simpler and sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We have also found that attempts to resum π2 terms using a timelike renormalization point have little numerical importance at N3LLp if the pveto T scale is around 20 to 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The production of a Higgs boson is an exception among single-boson processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In this case, the combination of larger corrections from color factors and slightly larger values of the scale (mH) appearing in the jet veto logarithms make resummation an important tool for improving the accuracy of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the appendix we have investigated the differences between the one-step and two-step procedures for calculating the hard function at the scale of pveto T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We find agreement within 2% of these two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The W +W − production process, where the jet veto has experimental importance, requires both resummation and matching to NNLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For the ZZ process resummation is mandatory but the matching to fixed order is less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Although this reflects the expectation that the resummed prediction is more accurate for systems of higher invariant mass, these findings depend on the exact nature of the cuts for each process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our work provides a comprehensive theoretical framework for studying jet vetoes in vector boson pair processes, and as data becomes available, a comparative experimental study would be of great interest and could help to validate our theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Acknowledgments RKE would like to thank Simone Alioli, Thomas Becher, Andrew Gilbert, Pier Monni and Philip Sommer for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In addition, RKE would like to thank TTP in Karlsruhe for hospitality during the drafting of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' TN would like to thank Robert Szafron for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' SS is supported in part by the SERB-MATRICS under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' MTR/2022/000135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This manuscript has been authored by Fermi Research Alliance, LLC under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' DE-AC02-07CH11359 with the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of High Energy Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This research used resources of the Wilson High- Performance Computing Facility at Fermilab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' DE-AC02-05CH11231 using NERSC award HEP-ERCAP0021890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 29 – A Reduced beam functions We have used the two loop beam function in the presence of a jet veto calculated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Their calculation, together with the corresponding soft function [25] has been performed in SCET using the exponential rapidity regulator [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The beam function for quark initiated processes in the presence of a jet veto has also been presented in Mellin space in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The calculation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [24] has a perturbative expansion, Iij = ∞ � k=0 �αs 4π �k I(k) ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) The beam functions with a jet veto are decomposed into a reference observable, the beam function for the transverse momentum of a color singlet observable and a remainder term accounting for the effects of jet clustering, Iij(x, Q, pveto T , R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µ, ν) = I⊥ ij(x, Q, pveto T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µ, ν) + ∆Iij(x, Q, pveto T , R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µ, ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) Since the divergence structure of the reference observable is the same as the beam function with a jet veto, ∆Iij can be calculated in four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Results for the reference observable are available in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The reduced beam function kernels ¯I as used in our setup are extracted from the coefficient I as ¯Iij(z, pveto T , R, µ) = e−hA(pveto T ,µ) Iij(z, pveto T , R, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) They similarly follow a perturbative expansion ¯Iik(z, pveto T , R, µ) = δik δ(1−z)+ αs 4π ¯I(1) ik (z, pveto T , µ)+ �αs 4π �2 ¯I(2) ik (z, pveto T , R, µ)+O(α3 s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) Contributions at order αs The αs contributions to ¯I were first obtained in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [3, 50] and read, ¯Iij(z, pveto T , R, µ) = δ(1 − z) δij + αs 4π � −2P (1) ij (z) L⊥ + R(1) ij (z) � + O(α2 s) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) where L⊥ = 2 ln(µ/pveto T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' R is the jet measure used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) and R(1)(z) is a remainder function given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' At this order there is no dependence on the jet radius, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Throughout this paper we expand in powers of αs/(4π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The one exception to this rule are the perturbative DGLAP splitting functions, Pij(z) = αs 2πP (1)(z) + �αs 2π �2 P (2)(z) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6) – 30 – Explicit expressions for P (1) and P (2) are given in Appendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The remainder functions at order αS are [66] R(1) qq (z) = CF � 2(1 − z) − π2 6 δ(1 − z) � , R(1) qg (z) = 4TF z(1 − z) , R(1) gg (z) = −CA π2 6 δ(1 − z) , R(1) gq (z) = 2CF z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7) where CA = 3, CF = 4 3, TF = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Contributions at order α2 s At order α2 s we have ¯I(2) ik (z, pveto T , R, µ) = � 2P (1) ij (x) ⊗ P (1) jk (y) − β0P (1) ik (z) � L2 ⊥ + � − 4P (2) ik (z) + β0R(1) ik (z) − 2R(1) ij (x) ⊗ P (1) jk (y) � L⊥ + R(2) ik (z, R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8) In this equation ⊗ represents a convolution, f(x) ⊗ g(y) = � 1 0 dx � 1 0 dyf(x) g(y) δ(z − xy) = � 1 z dy y f(z/y) g(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9) Explicit expressions for P (1) and P (2) are given in Appendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The expressions for P (1) ⊗ P (1), R(1) ⊗ P (1) are given in appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The results from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [24, 25] recast in the language of reduced beam functions allow us to extract R(2) ik (z, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We have checked that the reduced beam functions have the form predicted by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In addition, we have confirmed the known results for the α2 s R- dependent contribution to the collinear anomaly exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The result for the collinear anomaly exponent is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Structure of the two-loop reduced beam function While a numerical evaluation of the analytical formulas for the reduced beam functions is possible, we choose to perform a spline interpolation for improved numerical efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The reduced beam functions contain distributions of the following structure, ¯I(2) ij (z, pveto T , R, µ) = ¯I(2) ij,−1(pveto T , R, µ) δ(1 − z) + ¯I(2) ij,0(pveto T , R, µ) D0(1 − z) + ¯I(2) ij,1(pveto T , R, µ) D1(1 − z) + ¯I(2) ij,2(z, pveto T , R, µ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10) where, D0(1 − z) = 1 [1 − z]+ , D1(1 − z) = �ln(1 − z) (1 − z) � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11) ¯I(2) ij,2(z, pveto T , R, µ) contains terms which are regular at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 31 – The analytic results for the beam function of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [24] are presented as a power series in R up to powers of R8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The functions themselves contain powers of 1/(1 − z)n, in certain cases up to n = 7 or 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, these singularities at z = 1 are fictitious as can be seen by explicit expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The beam functions require special treatment in this region for numerical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The dominant region in the convolution of the function ¯I with the parton distributions is precisely the region z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' If we assume a parton distribution f(x) ∼ 1/x we have, ¯I ⊗ f = � 1 x dz z ¯I(z) f(x/z) ∼ 1 x � 1 x dz ¯I(z) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12) showing that all regions of z contribute equally to the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However if, as expected, the parton distribution function falls off more rapidly as x → 1, say f(x) ∼ (1 − x)n/x, ¯I ⊗ f = � 1 x dz z ¯I(z) f(x/z) ∼ 1 x � 1 x dz ¯I(z) (1 − x/z)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='13) Thus, it is precisely the large values of z which are crucial for the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In other words, the parton shower process is dominated by cascade from nearby values of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Larger cascades from more distant points are suppressed by the fall-off of the parton distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In view of the importance of the region z = 1, for numerical stability we perform an expansion about z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The absolute value of R(2) for the various parton transitions is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Individual R-dependent terms contain expressions of the form R2n/(1 − z)k where k can be a high power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, the singularity at z = 1 is only apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The resultant limiting forms obtained by series expansion about z = 1 are shown by the dashed lines in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In practice, we switch to the expanded form at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9, although the figures demonstrate that the expanded forms are accurate down to much smaller values of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B Definition of the beta function and anomalous dimensions The coefficients βn, ΓA n and γg n have perturbative expansions in powers of the renormalized coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Details are presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Expansion of β-function The beta function is defined as, dαs(µ) d ln µ = β(µ) = −2αs(µ) ∞ � n=0 βn �αs 4π �n+1 = −2αs(µ) αs(µ) 4π � β0 + β1 αs(µ) 4π + β2 �αs(µ) 4π �2 + β3 �αs(µ) 4π �3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) – 32 – (a) gg case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (b) qq case (c) gq case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (d) qg case (e) ¯qq case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (f) q′q case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Figure 18: Absolute value of R(2) for jet measure R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The ¯q′q case is the same as the q′q case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The sign of the contribution in the various regions is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The coefficients of the MS β function to four loops are [67–69],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' β0 = 11 3 CA − 4 3 TF nf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 33 – β1 = 34 3 C2 A − �20 3 CA + 4CF � TF nf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' β2 = 2857 54 C3 A + � C2 F − 205 18 CF CA − 1415 54 C2 A � 2TF nf + �11 9 CF + 79 54 CA � 4T 2 F n2 f ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='β3 = C4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�150653 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='486 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− 44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ C3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='ATF nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='−39143 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 136 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='ACF TF nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�7073 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='243 − 656 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ CAC2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F TF nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='−4204 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 352 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+46C3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F TF nf + C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='AT 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�7930 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 224 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F T 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�1352 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− 704 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+CACF T 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�17152 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='243 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 448 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 424 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='243CAT 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F n3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='f + 1232 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='243 CF T 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F n3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+dabcd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='dabcd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='NA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='−80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 + 704 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ nf ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) For the normalization of the SU(N) generators, the conventions of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [69, 70] are, dabcd A dabcd A NA = N2(N2 + 36) 24 , dabcd F dabcd A NA = N(N2 + 6) 48 , dabcd F dabcd F NA = N4 − 6N2 + 18 96N2 , NA = N2 − 1 , NF = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) Numerical values for the β-function coefficients are, β0 = 11 − 2 3 nf , β1 = 102 − 38 3 nf , β2 = 2857 2 − 5033 18 nf + 325 54 n2 f , β3 = 149753 6 + 3564ζ3 − �1078361 162 + 6508 27 ζ3 � nf + �50065 162 + 6472 81 ζ3 � n2 f + 1093 729 n3 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 Cusp Anomalous Dimension The cusp anomalous dimension depends on the label B which takes the two values, B = A, F for gluons and quarks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Its perturbative expansion is, ΓB cusp(µ) = ∞ � n=0 ΓB n �αs 4π �n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) – 34 – The coefficients up to four loops are [71, 72], ΓB 0 = 4CB , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6) ΓB 1 = 16CB � (CA �67 36 − π2 12 � − 5 9nfTF � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7) ΓB 2 = 64CB � C2 A �11ζ3 24 + 245 96 − 67π2 216 + 11π4 720 � + nfTF CF � ζ3 − 55 48 � + nfTF CA � −7ζ3 6 − 209 216 + 5π2 54 � − 1 27(nfTF )2 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='ΓB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 = 256CB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='C3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�1309ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− 11π2ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='16 − 451ζ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='288 + 42139 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10368 − 5525π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7776 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 451π4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5760 − 313π6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='90720 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ nfTF C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='−361ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 7π2ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 131ζ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− 24137 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10368 + 635π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1944 − 11π4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2160 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ nfTF CF CA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�29ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− π2ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 5ζ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 − 17033 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5184 + 55π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='288 − 11π4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='720 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ nfTF C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�37ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='24 − 5ζ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 + 143 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='288 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ (nfTF )2CA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�35ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='27 − 7π4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1080 − 19π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='972 + 923 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5184 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ (nfTF )2CF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='−10ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ π4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='180 + 299 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='648 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ (nfTF )3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='81 + 2ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 256dabcd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='dabcd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='NB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 − 3ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 55ζ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12 − π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12 − 31π6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7560 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 256nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='dabcd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='dabcd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='NB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 − ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 − 5ζ5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9) In addition to the relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) we need the related quantities, dabcd F dabcd A NF = (N2 − 1)(N2 + 6) 48 , dabcd F dabcd F NF = (N2 − 1)(N4 − 6N2 + 18) 96N3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 Non-cusp anomalous dimension The non-cusp anomalous dimension has the expansion, γq,g(µ) = ∞ � n=0 γq,g n �αs 4π �n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11) – 35 – We take the coefficients up to three loops from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [73] Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4, γq 0 = −3CF , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12) γq 1 = C2 F � 2π2 − 3 2 − 24ζ3 � + CF CA � 26ζ3 − 961 54 − 11π2 6 � + CF TF nf �130 27 + 2π2 3 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='13) γq 2 = C3 F � − 29 2 − 3π2 − 8π4 5 − 68ζ3 + 16π2 3 ζ3 + 240ζ5 � + C2 F CA � − 151 4 + 205π2 9 + 247π4 135 − 844 3 ζ3 − 8π2 3 ζ3 − 120ζ5 � + CF C2 A � − 139345 2916 − 7163π2 486 − 83π4 90 + 3526 9 ζ3 − 44π2 9 ζ3 − 136ζ5 � + C2 F TF nf �2953 27 − 26π2 9 − 28π4 27 + 512 9 ζ3 � + CF CATF nf � − 17318 729 + 2594π2 243 + 22π4 45 − 1928 27 ζ3 � + CF T 2 F n2 f �9668 729 − 40π2 27 − 32 27ζ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='14) From ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [74], Eq A5 we take, γg 0 = −β0 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15) γg 1 = C2 A �11π2 18 − 692 27 + 2ζ3 � + CATF nf �256 27 − 2π2 9 � + 4CF TF nf = C2 A � 2ζ3 − 59 9 � + CAβ0 �π2 6 − 19 9 � − β1 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='16) γg 2 = C3 A � − 97186 729 + 6109π2 486 − 319π4 270 + 122 3 ζ3 − 20π2 9 ζ3 − 16ζ5 � + C2 ATF nf �30715 729 − 1198π2 243 + 82π4 135 + 712 27 ζ3 � + CACF TF nf �2434 27 − 2π2 3 − 8π4 45 − 304 9 ζ3 � − 2C2 F TF nf + CAT 2 F n2 f � − 538 729 + 40π2 81 − 224 27 ζ3 � − 44 9 CF T 2 F n2 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='17) Primary references for the calculation of these coefficients can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We now present results for γS and γt which are needed for the implementation of the two-step calculation of the hard function for Higgs boson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [61] we have, for the first three expansion coefficients of the anomalous dimension γS that enters the evolution – 36 – equation of the hard matching coefficient CS (see also [59, 60]), γS 0 = 0 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18) γS 1 = C2 A � −160 27 + 11π2 9 + 4ζ3 � + CATF nf � −208 27 − 4π2 9 � − 8CF TF nf , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='19) γS 2 = C3 A �37045 729 + 6109π2 243 − 319π4 135 + �244 3 − 40π2 9 � ζ3 − 32ζ5 � + C2 ATF nf � −167800 729 − 2396π2 243 + 164π4 135 + 1424 27 ζ3 � + CACF TF nf �1178 27 − 4π2 3 − 16π4 45 − 608 9 ζ3 � + 8C2 F TF nf + CAT 2 F n2 f �24520 729 + 80π2 81 − 448 27 ζ3 � + 176 9 CF T 2 F n2 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='20) The function γt is given by, γt(αs) = α2 s d dαs � β(αs) α2s � = −2β1 �αs 4π �2 − 4β2 �αs 4π �3 − 6β3 �αs 4π �4 + O(α5 s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='21) As shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='22) µ independence provides the constraint, 2γg(αs) = γt(αs) + γS(αs) + β(αs)/αs , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='22) leading to the simple relationship between the coefficients in γg and γS, γS 0 = 2γg 0 + 2β0 , γS 1 = 2γg 1 + 4β1 , γS 2 = 2γg 2 + 6β2, γS 3 = 2γg 3 + 8β3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='23) C Definitions for beam function ingredients C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Exponent h We define the auxiliary functions hB for B = F, A which, when combined with the hard function and the collinear anomaly factor, will yield a renormalization group invariant hard function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' hF/A is defined to satisfy the RGE equation, d d ln µ hF/A(pveto T , µ) = 2 ΓF/A cusp(µ) ln µ pveto T − 2 γq/g(µ) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) The factor h removes logarithms from the beam function and has a perturbative expansion in terms of the renormalized coupling, hB(pveto T , µ) = αs 4πhB 0 + �αs 4π �2 hB 1 + �αs 4π �3 hB 2 + �αs 4π �4 hB 3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) – 37 – Thus for the particular case B = F we have that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' hF 0 (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µ) = 1 4ΓF 0 L2 ⊥ − γq 0L⊥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' hF 1 (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µ) = 1 12ΓF 0 β0L3 ⊥ + 1 4(ΓF 1 − 2γq 0β0)L2 ⊥ − γq 1L⊥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' hF 2 (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µ) = 1 24ΓF 0 β2 0L4 ⊥ + ( 1 12ΓF 0 β1 + 1 6ΓF 1 β0 − 1 3γq 0β2 0)L3 ⊥ + (1 4ΓF 2 − 1 2γq 0β1 − γq 1β0)L2 ⊥ − γq 2L⊥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' hF 3 (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µ) = + 1 40ΓF 0 β3 0L5 ⊥ + ( 5 48ΓF 0 β0β1 + 1 8ΓF 1 β2 0 − 1 4γq 0β3 0)L4 ⊥ + ( 1 12ΓF 0 β2 + 1 6ΓF 1 β1 + 1 4ΓF 2 β0 − 5 6γq 0β0β1 − γq 1β2 0)L3 ⊥ + (1 4ΓF 3 − 1 2γq 0β2 − γq 1β1 − 3 2γq 2β0)L2 ⊥ − γq 3L⊥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) where L⊥ = 2 ln(µ/pveto T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The corresponding result for B = A, q = g, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' for incoming gluons) is given by a similar expression mutatis mutandis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The expansion coefficients of the β-function, ΓF/A cusp and γq/g, used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3), are as given in Appendices B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 One loop splitting functions The one-loop DGLAP splitting functions as defined in [75] are P (1) qq (z) = CF �1 + z2 1 − z � + , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) P (1) qg (z) = TF � z2 + (1 − z)2� , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) P (1) gg (z) = 2CA � z (1 − z)+ + 1 − z z + z(1 − z) � + β0 2 δ(1 − z) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6) P (1) gq (z) = CF 1 + (1 − z)2 z , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 Two loop splitting functions Now we turn to the two-loop anomalous dimensions that contribute at sub-leading log level to the transitions between parton types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the quark sector there are four independent transitions that we must produce values for (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' q′ ← q,¯q′ ← q,q ← q and ¯q ← q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' They are expressed in terms of four functions, P (2) q′q = P S(2) qq , P (2) ¯q′q = P S(2) ¯qq , P (2) qq = P V (2) qq + P S(2) qq , P (2) ¯qq = P V (2) ¯qq + P S(2) ¯qq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8) – 38 – At next-to-leading order, the functions P S qq and P S ¯qq are non-zero, but we have the additional relation, P S qq = P S ¯qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' To facilitate the presentation we define the auxiliary functions, pqq(z) = 2 1 − z − 1 − z , p(r) qq (z) = −1 − z , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9) pqg(z) = z2 + (1 − z)2 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10) pgq(z) = 1 + (1 − z)2 z , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11) pgg(z) = 1 1 − z + 1 z − 2 + z(1 − z), p(r) gg (z) = 1 z − 2 + z(1 − z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12) The two valence functions needed for the quark sector are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [76–78],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' P V (2) qq (z) = C2 F � − � 2 ln z ln(1 − z) + 3 2 ln z � pqq(z) − � 3 2 + 7 2z � ln z − 1 2(1 + z) ln2 z − 5(1 − z) � +CF CA � (1 + z) ln z + 20 3 (1 − z) + � 1 2 ln2 z + 11 6 ln z � pqq(z) + � 67 18 − π2 6 �� 1 (1 − z)+ + p(r) qq (z) �� −CF TF nf � 4 3(1 − z) + 2 3pqq(z) ln z + 10 9 � 1 (1 − z)+ + p(r) qq (z) �� + � C2 F � 3 8 − π2 2 + 6ζ3 � + CF CA � 17 24 + 11π2 18 − 3ζ3 � −CF TF nf � 1 6 + 2π2 9 �� δ(1 − z) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='13) P V (2) ¯qq (z) = CF � CF − CA 2 �� 2pqq(−z)S2(z) + 2(1 + z) ln z + 4(1 − z) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='14) and for the singlet function we have, P S(2) qq = CF TF � 20 9z − 2 + 6z − 56 9 z2 + (1 + 5z + 8 3z2) ln z − (1 + z) ln2 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15) The other three transitions are simply given by, P (2) qg = CF TF � 2 − 9 2z − (1 2 − 2z) ln z − (1 2 − z) ln2 z + 2 ln(1 − z) – 39 – + � ln2 � 1 − z z � − 2 ln � 1 − z z � − π2 3 + 5 � pqg(z) � +CATF � 91 9 + 7 9z + 20 9z + � 68 3 z − 19 3 � ln z −2 ln(1 − z) − (1 + 4z) ln2 z + pqg(−z)S2(z) + � − 1 2 ln2 z + 22 3 ln z − ln2(1 − z) + 2 ln(1 − z) + π2 6 − 109 9 � pqg(z) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='16) P (2) gq (z) = C2 F � − 5 2 − 7z 2 + � 2 + 7 2z � ln z − � 1 − 1 2z � ln2 z − 2z ln(1 − z) − � 3 ln(1 − z) + ln2(1 − z) � pgq(z) � +CF CA � 28 9 + 65 18z + 44 9 z2 − � 12 + 5z + 8 3z2 � ln z +(4 + z) ln2 z + 2z ln(1 − z) + S2(z)pgq(−z) + � 1 2 − 2 ln z ln(1 − z) + 1 2 ln2 z + 11 3 ln(1 − z) + ln2(1 − z) − π2 6 � pgq(z) � +CF TF nf � − 4 3z − � 20 9 + 4 3 ln(1 − z) � pgq(z) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='P (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='gg (z) = CF TF nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− 16 + 8z + 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 z2 + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3z − (6 + 10z) ln z − (2 + 2z) ln2 z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+CATF nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 − 2z + 26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='z2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3(1 + z) ln z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='−20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='(1 − z)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ p(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='gg (z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 (1 − z) + 67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='z2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 − 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 z + 44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 z2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='ln z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+4(1 + z) ln2 z + 2pgg(−z)S2(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='ln2 z − 4 ln z ln(1 − z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='pgg(z) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9 − π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='(1 − z)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ p(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='gg (z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 + 3ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− CF TF nf − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3CATF nf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='δ(1 − z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18) – 40 – The function S2(z) is defined by S2(z) = � 1 1+z z 1+z dy y ln �1 − y y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='19) In terms of the dilogarithm function Li2(z) = − � z 0 dy y ln(1 − y) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='20) we have S2(z) = −2 Li2(−z) + 1 2 ln2 z − 2 ln z ln(1 + z) − π2 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='21) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4 P (1) ⊗ P (1) and R(1) ⊗ P (1) We give here expressions for the convolutions of functions appearing in the beam functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The convolutions are defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Similar expressions have been given in [1, 12] The convolutions of the one-loop DGLAP kernels from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) are, P (1) qq ⊗ P (1) qg = CF TF � 2z − 1 2 + (2z − 4z2 − 1) ln z + (2 − 4z(1 − z)) ln(1 − z) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='22) P (1) qg ⊗ P (1) gg = CATF � 2(1 + 4z) ln z + 4 3z + 1 + 8z − 31 3 z2� + � 2CA ln(1 − z) + β0 2 � P (1) qg (z) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='23) P (1) gq ⊗ P (1) qq = C2 F � 2 − 1 2z + (2 − z) ln z � + 2CF P (1) gq (z) ln(1 − z) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='24) P (1) gg ⊗ P (1) gq = CACF � 8 + z + (4z3 − 31) 3z − 4(1 + z + z2) z ln z � + � 2CA ln(1 − z) + β0 2 � P (1) gq (z) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='25) P (1) qg ⊗ P (1) gq = CF TF � 2(1 + z) ln z + 1 − z + 4 3 (1 − z3) z � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='26) P (1) qq ⊗ P (1) qq = C2 F � 8 �ln(1 − z) (1 − z) � + − 4(1 + z) ln(1 − z) − 2(1 − z) + � 3 + 3z − 4 (1 − z) � ln z � + 3CF P (1) qq (z) − C2 F (9 4 + 4ζ2)δ(1 − z) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='27) P (1) gg ⊗ P (1) gg = 4C2 A � 2 �ln(1 − z) (1 − z) � + + 2((1 − z) z + z(1 − z) − 1) ln(1 − z) + 3(1 − z) − ( 1 1 − z + 1 z − z2 + 3z) ln z − 11(1 − z3) 3z � + β0P (1) gg (z) − (β2 0 4 + 4C2 Aζ2)δ(1 − z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='28) The convolutions of lowest order DGLAP kernels, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) with the one-loop finite terms in the beam functions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7) are, R(1) gg ⊗ P (1) gg = −CAζ2P (1) gg (z) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='29) – 41 – R(1) gq ⊗ P (1) qg = 2CF TF � (1 − z)(1 + 2z) + 2z ln z � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='30) R(1) qq ⊗ P (1) qq = CF � CF (1 − z)(4 ln(1 − z) − 2 ln z − 1) − ζ2P (1) qq (z) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='31) R(1) qg ⊗ P (1) gq = −4CF TF � 1 + z ln z − (1 + 2z3) 3z � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='32) R(1) qg ⊗ P (1) gg = −CATF (16z ln z − 68 3 z2 + 20z + 4 − 4 3z ) + (2CA ln(1 − z) + β0 2 )R(1) qg (z) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='33) R(1) qq ⊗ P (1) qg = CF TF (2z2 + 2z − 4 − (2 + 4z) ln z) − CF ζ2P (1) qg (z) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='34) R(1) gq ⊗ P (1) qq = −C2 F (2z ln z − 4z ln(1 − z) − z − 2) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='35) R(1) gg ⊗ P (1) gq = −CAζ2P (1) gq (z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='36) D Rapidity anomalous dimension Solving the collinear anomaly RG equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='13)) as an expansion in αs (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15)) we have that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F (0) gg (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µh) = ΓA 0 L⊥ + dveto 1 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F (1) gg (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µh) = 1 2ΓA 0 β0L2 ⊥ + ΓA 1 L⊥ + dveto 2 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F (2) gg (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µh) = 1 3ΓA 0 β2 0L3 ⊥ + 1 2(ΓA 0 β1 + 2ΓA 1 β0)L2 ⊥ + (ΓA 2 + 2β0dveto 2 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A))L⊥ + dveto 3 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F (3) gg (pveto T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' µh) = 1 4β3 0ΓA 0 L4 ⊥ + (ΓA 1 β2 0 + 5 6ΓA 0 β0β1)L3 ⊥ + (1 2ΓA 0 β2 + ΓA 1 β1 + 3 2ΓA 2 β0 + 3dveto 2 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A)β2 0)L2 ⊥ + (ΓA 3 + 3dveto 3 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A)β0 + 2dveto 2 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A)β1)L⊥ + dveto 4 (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) where L⊥ = 2 ln(µh/pveto T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The corresponding result for Fqq is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Because Fgg appears in the exponent, we see that dveto 1 contributes in NLL, dveto 2 in NNLL, and dveto 3 in N3LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 42 – D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 dveto 2 expansion The expansion coefficients for dveto 2 , which is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18), are given by [4, 5, 12], cA L = 131 72 − π2 6 − 11 6 ln 2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='096259 , cA 0 = −805 216 + 11π2 72 + 35 18 ln 2 + 11 6 ln2 2 + ζ3 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6106495 , cA 2 = 1429 172800 + π2 48 + 13 180 ln 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='263947 , cA 4 = − 9383279 406425600 − π2 3456 + 587 120960 ln 2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0225794 , cA 6 = 74801417 97542144000 − 23 67200 ln 2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='29625 · 10−4 , cA 8 = − 50937246539 2266099089408000 − π2 24883200 + 28529 1916006400 ln 2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='25537 · 10−5 , cA 10 = 348989849431 243708656615424000 − 3509 3962649600 ln 2 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18201 · 10−7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) and cf L = −23 36 + 2 3 ln 2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1767908 , cf 0 = 157 108 − π2 18 − 8 9 ln 2 − 2 3 ln2 2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='03104049 , cf 2 = 3071 86400 − 7 360 ln 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0220661 , cf 4 = − 168401 101606400 + 53 30240 ln 2 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='42544 · 10−4 , cf 6 = 7001023 48771072000 − 11 100800 ln 2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='79076 · 10−5 , cf 8 = − 5664846191 566524772352000 + 4001 479001600 ln 2 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='20958 · 10−6 , cf 10 = 68089272001 83774850711552000 − 13817 21794572800 ln 2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='73334 · 10−7 , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) We see that for values of the jet radius R < 1 the terms c6, c8 and c10 can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For the gluon case the expansion of the function in numerical form is, f(R, A) = − (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0963 CA + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1768 TF nf) ln R + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6106 CA − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0310 TF nf) + (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5585 CA + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0221 TF nf) R2 + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0399 CA − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0004 TF nf) R4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) whereas for the quark case we have f(R, F) = − (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0963 CA + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1768 TF nf) ln R + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6106 CA − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0310 TF nf) + (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8225 CF + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2639 CA + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0221 TF nf) R2 + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0625 CF − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='02258 CA − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0004 TF nf) R4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) – 43 – E Renormalization Group Evolution The evolution equation matching for a generic hard matching coefficient C has the form, d d ln µ ln C(Q2, µ) = � Γcusp(αs(µ)) ln Q2 µ2 + γ(αs(µ)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) Following ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [26] the solution to the evolution equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) is, C(Q2, µ) = exp [2S(µh, µ) − aγ(µh, µ)] �Q2 µ2 h �−aΓ(µh,µ) C(Q2, µh) , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) ln C(Q2, µ) = 2S(µh, µ) − aγ(µh, µ) − aΓ(µh, µ) ln �Q2 µ2 h � + ln C(Q2, µh) , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) where µh ∼ Q is a hard matching scale at which the Wilson coefficient C is calculated using fixed-order perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The Sudakov exponent S and the exponents aγ, aΓ are the solutions to the auxiliary differential equations, d d ln µ S(ν, µ) = −Γcusp � αs(µ) � ln µ ν , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) d d ln µ aΓ(ν, µ) = −Γcusp � αs(µ) � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) d d ln µ aγ(ν, µ) = −γ � αs(µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6) with the boundary conditions S(ν, ν) = aΓ(ν, ν) = aγ(ν, ν) = 0 at µ = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) we recover Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The solutions to the evolution equation are conveniently expressed in terms of the running coupling, aΓ(ν, µ) = − αs(µ) � αs(ν) dα Γcusp(α) β(α) , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7) S(ν, µ) = − αs(µ) � αs(ν) dα Γcusp(α) β(α) α � αs(ν) dα′ β(α′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8) Substituting the values for the beta function coefficients in the MS scheme given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 and the values for cusp anomalous dimension given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7) we obtain, aΓ(µh, µ) = aΓ 0 + aΓ 1 + aΓ 2 + aΓ 3 , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9) – 44 – where the coefficients in the expansion are, aΓ 0 = Γ0 ln(r) 2β0 , r = αs(µ)/αs(µh) , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10) aΓ 1 = αs(µh)(r − 1)(β0Γ1 − β1Γ0) 8πβ2 0 , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11) aΓ 2 = α2 s(µh)(r2 − 1) � −β0β1Γ1 + β0(β0Γ2 − β2Γ0) + β2 1Γ0 � 64π2β3 0 , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12) aΓ 3 = −α3 s(µh) � r3 − 1 � × � β2 0(−β0Γ3 + β2Γ1 + β3Γ0) − β0β2 1Γ1 + β0β1(β0Γ2 − 2β2Γ0) + β3 1Γ0 � 384π3β4 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='13) The solution for aγ follows from the one for aΓ by making the replacement Γk → γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The non-cusp anomalous dimensions γ are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Evaluating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8) to obtain the evolution for S we get, S(µh, µ) = S0 + S1 + S2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='14) with, S0 = 1 8β3 0 � 8πβ0Γ0(r + r(− ln(r)) − 1) αs(µh)r + 2(r − 1)(β1Γ0 − β0Γ1) + ln(r)(2β0Γ1 + β1Γ0 ln(r) − 2β1Γ0) � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15) S1 = −αs(µh) 32πβ4 0 � 2 ln(r) � −β0β1Γ1r + β0β2Γ0 + β2 1Γ0(r − 1) � + (r − 1) � −β0β1Γ1(r − 3) + β0(β0(r − 1)Γ2 − β2Γ0(r + 1)) + β2 1Γ0(r − 1) � � ,(E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='16) S2 = α2 s(µh) 256π2β5 0 � 2 ln(r) � β1r2 � −β0β1Γ1 + β0(β0Γ2 − β2Γ0) + β2 1Γ0 � − Γ0 � β2 0β3 − 2β0β1β2 + β3 1 � � + (r − 1) � β2 0(2(β0(r + 1)Γ3 − 2β2Γ1) − β3Γ0(r + 1)) + β0β2 1Γ1(r + 5) + β0β1(β2Γ0(r + 5) − 3β0(r + 1)Γ2) − 4β3 1Γ0 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='17) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Recovery of the double log formula As we have seen S satisfies a RGE given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) with a solution given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The leading term in S0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15) is S0 ≈ πΓ0 β2 0αs(µh) � 1 + ln �1 r � − 1 r � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18) – 45 – where r = αs(µ)/αs(µh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In this form the presence of a double log is obscured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We can easily recover the double log by retaining only the leading terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The leading expression for r is given by solving the equation for the beta function, 1 r = 1 − αs(µh) 2π β0 ln �µh µ � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='19) S0 ≈ πΓ0 β2 0αs(µh) �αs(µh) 2π β0 ln �µh µ � + ln � 1 − αs(µh) 2π β0 ln �µh µ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='20) Expanding for small αs(µh) ln(µh/µ) we get, S(µh, µ) ≈ −Γ0 2 αS(µh) 4π ln2 �µh µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='21) This gives the expected log squared with a negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' F The hard function for the Drell-Yan process The form factors of the vector current have been presented several places in the literature [79–84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The bare form factor is given as, F q,bare(q2, µ2) = 1 + �αbare s 4π � (∆)ϵFq 1 + �αbare s 4π �2 (∆)2ϵFq 2 + O(α3 s) , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) where, ∆ = 4πe−γE � µ2 −q2 − i0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) In the following we will drop 4πe−γE, so that all poles should be understood in the MS sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The values found for the bare coefficients are, Fq 1 = CF � − 2 ϵ2 − 3 ϵ + ζ2 − 8 + ϵ �3ζ2 2 + 14ζ3 3 − 16 � + ϵ2 �47ζ2 2 20 + 4ζ2 + 7ζ3 − 32 �� + O(ϵ3) , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) Fq 2 = C2 F � 2 ϵ4 + 6 ϵ3 − 1 ϵ2 � 2ζ2 − 41 2 � − 1 ϵ �64ζ3 3 − 221 4 � − � 13ζ2 2 − 17ζ2 2 + 58ζ3 − 1151 8 �� + CF CA � − 11 6ϵ3 + 1 ϵ2 � ζ2 − 83 9 � − 1 ϵ �11ζ2 6 − 13ζ3 + 4129 108 � + �44ζ2 2 5 − 119ζ2 9 + 467ζ3 9 − 89173 648 �� + CF nf � 1 3ϵ3 + 14 9ϵ2 + 1 ϵ �ζ2 3 + 353 54 � + �14ζ2 9 − 26ζ3 9 + 7541 324 �� + O(ϵ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) – 46 – The renormalized form factor can then be written as, F q(µ2, q2, ϵ) = 1 + �αs(µ) 4π � F q 1 (µ2, q2, ϵ) + �αs(µ) 4π �2 F q 2 (µ2, q2, ϵ) + O(α3 s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) where, F q 1 (µ2, q2, ϵ) = ∆ϵFq 1 , F q 2 (µ2, q2, ϵ) = ∆2ϵFq 2 − β0 ϵ ∆ϵFq 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6) In the full theory the matrix element between on-shell massless quark and gluon states, after charge renormalization is given by F q(µ2, q2, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Charge renormalization has removed the UV poles, but the renormalized form factor still contains IR poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The matrix element in the effective theory involves only scaleless, dimensionally regulated integrals and hence is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This vanishing can be interpreted as a cancellation between ultra-violet and infrared poles: 1 ϵIR − 1 ϵUV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7) After matching, the IR poles in the on-shell matrix element are effectively transformed into UV poles and need to be renormalized as follows, CV (αs(µ2), µ2, q2) = lim ϵ→0 � ZV (ϵ, µ2q2) �−1 F q(µ2, q2, ϵ) , ln � CV (αs(µ2), µ2, q2) � = ln � Fq(µ2, q2, ϵ) � − ln � ZV (ϵ, µ2, q2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8) The renormalization constant, ZV contains only pure pole terms, ln ZV (ϵ, µ2, q2) = �αs 4π � � − ΓF 0 2ϵ2 + 1 2ϵ � ΓF 0 L + 2γq 0 �� + �αs 4π �2 � 3ΓF 0 β0 8ϵ3 − 1 ϵ2 �ΓF 0 β0 4 L − CF � CA(16 9 + ζ2 � + 4 9nf) � + 1 4ϵ � ΓF 1 L + 2γq 1 �� , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9) where L = ln((−q2 − i0)/µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The matching coefficients have a perturbative expansion in terms of the renormalized cou- pling, CV (αs(µ2), µ2, q2) = 1 + ∞ � n=1 �αs(µ2) 4π �n CV n (µ2, q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10) The matching coefficients, which are known to two loop order [85, 86] (and beyond [84]) for Drell-Yan production, can be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8): CV 1 = CF � − L2 + 3L − 8 + ζ2 � , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11) – 47 – CV 2 = C2 F �1 2L4 − 3L3 + �25 2 − ζ2 � L2 + � − 45 2 + 24ζ3 − 9ζ2 � L + 255 8 − 30ζ3 + 21ζ2 − 83 10ζ2 2 � +CF CA �11 9 L3 + � − 233 18 + 2ζ2 � L2 + �2545 54 − 26ζ3 + 22 3 ζ2 � L − 51157 648 + 313 9 ζ3 − 337 18 ζ2 + 44 5 ζ2 2 � + CF nf � − 2 9L3 + 19 9 L2 + � − 209 27 − 4 3ζ2 � L + 4085 324 + 2 9ζ3 + 23 9 ζ2 � , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12) where L = ln((−q2 − i0)/µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' CV satisfies the renormalization group equation, d d ln µ ln[CV (αs(µ2), µ2, q2)] = ΓF cusp(µ) ln �−q2 − i0 µ2 � + 2γq(µ) , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='13) with the anomalous dimensions as given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 and Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The derivation of the hard function for boson pair processes has been described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' G The hard function for Higgs production G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Implementation of one-step procedure The one-step procedure [1, 13] is based on the observation that the ratio mt/mH is not large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For an on-shell Higgs boson the parameter, m2 H/m2 t ≈ 1 2 whereas αs ln(m2 t /m2 H) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='65αs, indicating that power corrections should be more important than resumming logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The matching is performed at a scale µh by integrating out the top quark and all gluons and light quarks with off-shellness above µh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The hard Wilson coefficient so defined satisfies the RGE, µ d dµ ln CH(m2 t , q2, µ2) = ΓA cusp(αs(µ)) ln −q2 − i0 µ2 + 2γg[αs(µ)] , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) where Γcusp and γg are given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' As a consequence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) the Wilson coefficient has the following structure, CH(m2 t , q2, µ2 h) = αs(µh)F H 0 � q2 4m2 t �� 1 + αs(µh) 4π � CH 1 �−q2 − i0 µ2 h � + F H 1 � q2 4m2 t �� + � αs(µh) (4π) �2� CH 2 �−q2 − i0 µ2 h , q2 4m2 t � + F H 2 � q2 4m2 t ��� , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) The finite terms can be derived from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [88], F H 0 (z) = 3 2z − 3 2z ���1 − 1 z ��� � arcsin2(√z) , 0 < z ≤ 1 , ln2[−i(√z + √z − 1)] , z > 1 , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3) – 48 – ≈ 1 + 7z 30 + 2z2 21 + 26z3 525 + 512z4 17325 + O(z5), z < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) For the values of mt and mH in Table 2, |F H 0 (z0)|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0653 , z0 = m2 H 4m2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5) The coefficients CH 1 and CH 2 are fixed by the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' CH 1 (L) = CA � −L2 + π2 6 � , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6) CH 2 (L, z) = 1 2C2 AL4 + 1 3CAβ0L3 + CA �� −4 3 + π2 6 � CA − 5 3β0 − F1(z) � L2 + ��59 9 − 2ζ3 � C2 A + �19 9 − π2 3 � CAβ0 − F1(z)β0 � L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7) where z = q2/4/m2 t and L = ln[(−q2 − i0)/µ2 h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The full analytic mt dependence of the virtual two-loop corrections to gg → H in terms of harmonic polylogarithms were obtained in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [89–91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' For our purposes the results expanded in m2 H/m2 t from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [88, 92, 93] will be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The functions F H 1 (z), F H 2 (z) which, together with F H 0 (z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4) encode the mt dependence of the hard Wilson coefficient in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Following the procedure described in Appendix F they are easily extracted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [88], F H 1 (z) = � 5 − 38 45 z − 1289 4725 z2 − 155 1134 z3 − 5385047 65488500 z4� CA + � −3 + 307 90 z + 25813 18900 z2 + 3055907 3969000 z3 + 659504801 1309770000 z4� CF + O(z5) (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 (z) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A + 11CACF − 6CF β0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='ln(−4z − i0) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='−419 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='27 + 7π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ π4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='72 − 44ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='−217 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 + 44ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='CACF + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�2255 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='108 + 5π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12 + 23ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='CAβ0 − 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6CATF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 − 12ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='CF β0 − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3CF TF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�11723 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='384 ζ3 − 404063 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='14400 − 223 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='108 ln(−4z − i0) − 19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='135π2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ CF CA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�2297 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='16 ζ3 − 1099453 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− 242 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='135 ln(−4z − i0) − 953 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='540π2 + 28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15π2 ln 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�13321 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='ζ3 − 36803 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='240 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3π2 − 56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15π2 ln 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ CF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�77 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12ζ3 − 4393 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='405 − 7337 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2700β0 + 39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10 ln(−4z − i0)β0 + 28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='45π2 + 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15π2β0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ CA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� 77 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='384ζ3 − 64097 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='129600 − 269 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='75 β0 + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15 ln(−4z − i0) − 31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='180 ln(−4z − i0)β0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ z2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�110251 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9216 ζ3 − 3084463261 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='254016000 − 2869 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4536 ln(−4z − i0) − 1289 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='28350π2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='– 49 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ CF CA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�2997917 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='23040 ζ3 − 55535378557 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='381024000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='− 18337 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='28350 ln(−4z − i0) − 128447 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='113400π2 + 1714 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1575π2 ln 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�36173 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='192 ζ3 − 95081911 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='453600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 857 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='630π2 − 3428 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1575π2 ln 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ CA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� 265053121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1524096000 − 16177 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='92160ζ3 − 45617 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='47250β0 + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='315 ln(−4z − i0) − 623 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5400 ln(−4z − i0)β0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ CF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�21973 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='7680 ζ3 − 8108339 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1555200 − 509813 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3969000β0 − 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15 ln(−4z − i0) + 29147 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18900 ln(−4z − i0)β0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ 1714 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4725π2 + 857 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3150π2β0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='+ O(z3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9) We can assess the quality of the expansion in z by numerical evaluation, CH(m2 t , q2, q2) = αs(q)F0(z) � 1 + 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9348αs 4π(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0158(8z) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00098312(8z)2) + 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0371 �αs 4π �2 (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1883(8z) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0120(8z)2) + 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='466 �αs 4π �2 ln(−8z − i0) π (1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0288(8z) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='001462(8z)2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10) In the vicinity of the Higgs boson pole (8z ≈ 1) subsequent terms in the z expansion are expected to contribute below the percent level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 Implementation of the two-step procedure In the two-step procedure of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [59–62] one first integrates out the top quark at a scale µt ≊ mt and subsequently matches from the QCD effective Lagrangian onto SCET at µh ≊ mH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Running between µh and µt allows one to sum logarithms of mt/mH, but one neglects power of mH/mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1 Ct(m2 t , µ2 t ) For a heavy top quark the effective Lagrangian for the production of a top quark is given by, Leff = Ct(m2 t , µ2 t ) H v αs(µ2 t ) 12π Gµν aGµν a , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11) where v ≈ 246 GeV is the Higgs boson vacuum expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The hard matching scale µt at which the Wilson coefficient can be computed perturbatively is of order mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The short distance coefficient Ct(m2 t , µ2) obeys the RGE, d d ln µCt(m2 t , µ2) = γt(αs) Ct(m2 t , µ2), γt(αs) = α2 s d dαs �β(αs) α2s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12) The expressions for the short-distance coefficient Ct(m2 t , µ2 t ) at NNLO is, Ct(m2 t , µ2 t ) = 1 + αs(µt) 4π Ct 1 + �αs(µt) 4π �2 Ct 2(m2 t , µ2 t ) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='13) where (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (12) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [61]), Ct 1 = 5CA − 3CF – 50 – Ct 2(m2 t , µ2 t ) = 27 2 C2 F + � 11 ln m2 t µ2 t − 100 3 � CF CA − � 7 ln m2 t µ2 t − 1063 36 � C2 A −4 3CF TF − 5 6CATF − � 8 ln m2 t µ2 t + 5 � CF TF nf − 47 9 CATF nf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='14) The evolution of these coefficients to the resummation scale µ is described in Appendix A of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The solution to the evolution equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12) for Ct at scale µ is, Ct(m2 t , µ2) = β(αs(µ)) α2s(µ) α2 s(µt) β(αs(µt)) Ct(m2 t , µ2 t ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='15) The result at NNLO for the square of the coefficient function is, � Ct(m2 t , µ2) �2 = 1 + �αs 4π �� 2Ct 1 + 2(rt − 1)β1 β0 � + �αs 4π �2� (Ct 1)2 + 2Ct 2(m2 t , µ2 t ) + (2β2β0 + β2 1) β2 0 (rt − 1)2 + 2(2β2β0 + 2β1β0Ct 1 − β2 1) β2 0 (rt − 1) � , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='16) where rt = αs(µ)/αs(µt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This extends the NLO result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (2) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2 CS(−q2, µh) CS is the Wilson coefficient matching the two gluon operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11) to an operator in SCET in which all the hard modes have been integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The result for the matching coefficient CS from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (16) and (17) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' It is given by, CS(−q2, µ2 h) = 1 + ∞ � n=1 CS n (L) �αs(µ2 h) 4π �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='17) The coefficient CS obeys the renormalization equation, d d ln µ CS(−q2 − iϵ, µ2) = � ΓA cusp(αs) ln −q2 − iϵ µ2 + γS(αs) � CS(−q2 − iϵ, µ2) , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18) with L = ln(−q2 − i0)/µ2 h and γS is given in Eq (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The logarithmic terms are determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The full results for the one- and two-loop coefficients are, CS 1 = CA � − L2 + π2 6 � , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='19) CS 2 = C2 A �L4 2 + 11 9 L3 + � − 67 9 + π2 6 � L2 + �80 27 − 11π2 9 − 2ζ3 � L + 5105 162 + 67π2 36 + π4 72 − 143 9 ζ3 � + CF TF nf � 4L − 67 3 + 16ζ3 � – 51 – + CATF nf � − 4 9 L3 + 20 9 L2 + �104 27 + 4π2 9 � L − 1832 81 − 5π2 9 − 92 9 ζ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='20) The full result for the renormalization group invariant hard function in the two-step scheme is, ¯H(mt, mH, pveto T ) = � αs(µ) αs(pveto T ) �2 (Ct(m2 t , µ))2 ��CS(−m2 H, µ) ��2 × � mH pveto T �−2Fgg(pveto T ,µ) e2hA(pveto T ,µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='21) The µ-independence of this hard function can be used to constrain γS, d d ln µ ¯H(mt, mH, pveto T ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='22) Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='13,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1) we can derive the relation between the collinear anomalous dimensions, 2γg(αs) = γt(αs) + γS(αs) + β(αs)/αs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='23) This relation could be cast in a more transparent form by noting that the quantity (αsCS) obeys a similar evolution equation to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='18), d d ln µ � αs(µ)CS(−m2 H − iϵ, µ2) � = αs(µ) � ΓA cusp(αs) ln −m2 H − iϵ µ2 + γS(αs) � CS(−m2 H − iϵ, µ2) + β(αs)CS(−m2 H − iϵ, µ2) = � ΓA cusp(αs) ln −m2 H − iϵ µ2 + γS′(αs) � � αs(µ)CS(−m2 H − iϵ, µ2) � , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='24) but with anomalous dimension γS′(αs) = γS(αs) + β(αs)/αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We then have the relation 2γg(αs) = γt(αs) + γS′(αs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This indicates that after the second matching, the evolution down to a lower scale satisfies the same renormalization equation in both the one-step and the two-step schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3 Assessment of the two schemes for the Higgs hard function The two schemes for the calculation of the hard function have application in jet veto resum- mation but also in the resummation of the Higgs boson transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A complete discussion of the error budget for Higgs boson production including scale dependence, parton distribution dependence, the influence of loops of b-quarks and electroweak corrections is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Here we shall simply compare and contrast the one-step and the two-step scheme, in the Higgs on shell region where m2 H ≈ m2 t /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' It is easy to check the internal consistency of the two schemes in the limit where we drop terms of order q2/(4m2 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Setting z = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2) and evaluating all coefficient functions at – 52 – a common scale µ, we have that, αs(µ) Ct(m2 t , µ2) CS(−q2, µ2) = CH(m2 t , q2, µ2)z=0 + O(α4 s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='25) We can test this equivalence numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We start by fixing µ2 = q2 and consider the quantities that enter the calculation of the cross-section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' the square of the absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the two-step scheme we have, |Ct(m2 t , q2)|2 = 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1957 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0204 , |Cs(−q2, q2)|2 = 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6146 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2155 , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='26) where the second and third terms represent the O(αs) and O(α2 s) terms respectively, evaluated using αs(q2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' In the one-step case we get, |CH z=0(m2 t , q2, q2)/αs(q)|2 = 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8104 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3563 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='27) Performing a strict fixed-order truncation of the product of the two-step result we have, � |Ct(m2 t , q2)|2|Cs(−q2, q2)|2� expanded = 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8104 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3563 , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='28) which is in perfect agreement with the one-step case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This indicates that the numerical implementation of the two procedures is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' If we instead evaluate the product after the individual expansions have been performed, a choice of equal formal accuracy, we have, |Ct(m2 t , q2)|2 expanded |Cs(−q2, q2)|2 expanded = 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='9306 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2953 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='29) This results in a significant difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We therefore work with with the strict fixed-order truncation throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We now restore the z-dependence in F H 1 and F H 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2), but still keep z = 0 in the overall factor F H 0 (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' We then find that the ratio of the one-step to the two-step becomes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0028 at NLO and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0053 at NNLO, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' these corrections are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Now we allow the matching scale for the top quark, µt to take its natural value, µt = mt and find one/two-step ratios of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0054 at NLO and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0073 at NNLO, again a small effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Finally, we reinstate the hard evolution down to the resummation scale and find that the ratio of the one-step to the two-step (at pveto T = 25 GeV) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0177 at NLO and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0125 at NNLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The cumulative effect at this point is noticeable but still small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' However, we note that we have so far kept z = 0 in the overall factor F H 0 (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' The one-step procedure is recovered by re-instating F H 0 (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' This implies that, in order to obtain the level of agreement quoted above between the two schemes, the overall factor of F H 0 (z) must also be applied to give a modified version of the two-step scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neglecting this step would result in a significant difference, since |F H 0 (z)|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0653 see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Our overall conclusion on the two schemes is in line with the known result that Higgs boson production has substantial corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Accounting for the most important mass effects by – 53 – rescaling the two-step result by the exact result at leading order, the one-step procedure gives a larger result than the two-step procedure for pveto T = 25 GeV at the level of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Any substantial difference between the two methods beyond this level is most likely due to uncontrolled higher order effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Berger, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Marcantonini, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Stewart, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Tackmann and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Waalewijn, Higgs Production with a Central Jet Veto at NNLL+NNLO, JHEP 04 (2011) 092 [1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4480].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Stewart, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Tackmann and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Waalewijn, N-Jettiness: An Inclusive Event Shape to Veto Jets, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 105 (2010) 092002 [1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2489].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neubert, Factorization and NNLL Resummation for Higgs Production with a Jet Veto, JHEP 07 (2012) 108 [1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3806].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Banfi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Salam and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zanderighi, NLL+NNLO predictions for jet-veto efficiencies in Higgs-boson and Drell-Yan production, JHEP 06 (2012) 159 [1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5773].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Banfi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Monni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Salam and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zanderighi, Higgs and Z-boson production with a jet veto, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 109 (2012) 202001 [1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4998].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Kallweit, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Re, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rottoli and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Wiesemann, Accurate single- and double-differential resummation of colour-singlet processes with MATRIX+RADISH: W +W − production at the LHC, JHEP 12 (2020) 147 [2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='07720].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Re, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rottoli and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Torrielli, Fiducial Higgs and Drell-Yan distributions at N3LL′+NNLO with RadISH, 2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='07509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Frederix, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neubert and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rothen, Automated NNLL + NLO resummation for jet-veto cross sections, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C 75 (2015) 154 [1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='8408].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Banfi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Monni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Salam and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zanderighi, “JetVHeto.” https://jetvheto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='hepforge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='org/, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Arpino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Banfi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Jäger and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Kauer, “MCFM-RE.” https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='com/lcarpino/MCFM-RE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Stefan Kallweit, Emanuele Re and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Wiesemann, “MCFM-RE.” https://matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='hepforge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='org/matrix+radish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='html, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neubert and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rothen, Factorization and N 3LLp+NNLO predictions for the Higgs cross section with a jet veto, JHEP 10 (2013) 125 [1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0025].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [13] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Stewart, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Tackmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Walsh and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zuberi, Jet pT resummation in Higgs production at NNLL′ + NNLO, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' D 89 (2014) 054001 [1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1808].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Banfi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Caola, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Dreyer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Monni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Salam, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zanderighi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=', Jet-vetoed Higgs cross section in gluon fusion at N3LO+NNLL with small-R resummation, JHEP 04 (2016) 049 [1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='02886].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Dawson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Jaiswal, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ramani and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zeng, Resummation of jet veto logarithms at N3LLa + NNLO for W +W − production at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' D 94 (2016) 114014 [1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='01034].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Arpino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Banfi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Jäger and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Kauer, BSM WW production with a jet veto, JHEP 08 (2019) 076 [1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='06646].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 54 – [17] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Li and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Liu, Resummation prediction on gauge boson pair production with a jet veto, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' D 93 (2016) 094020 [1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00509].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Salam, Towards Jetography, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C 67 (2010) 637 [0906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1833].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Cacciari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Salam and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Soyez, The anti-kt jet clustering algorithm, JHEP 04 (2008) 063 [0802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [20] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Dokshitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Leder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Moretti and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Webber, Better jet clustering algorithms, JHEP 08 (1997) 001 [hep-ph/9707323].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Wobisch and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Wengler, Hadronization corrections to jet cross-sections in deep inelastic scattering, in Workshop on Monte Carlo Generators for HERA Physics (Plenary Starting Meeting), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 270–279, 4, 1998 [hep-ph/9907280].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Catani, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Dokshitzer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Seymour and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Webber, Longitudinally invariant Kt clustering algorithms for hadron hadron collisions, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 406 (1993) 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ellis and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Soper, Successive combination jet algorithm for hadron collisions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' D 48 (1993) 3160 [hep-ph/9305266].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Abreu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Gaunt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Monni, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rottoli and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Szafron, Quark and gluon two-loop beam functions for leading-jet pT and slicing at NNLO, 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='07037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Abreu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Gaunt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Monni and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Szafron, The analytic two-loop soft function for leading-jet pT , 2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='02987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [26] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neubert and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Pecjak, Factorization and Momentum-Space Resummation in Deep-Inelastic Scattering, JHEP 01 (2007) 076 [hep-ph/0607228].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [27] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neill and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zhu, An exponential regulator for rapidity divergences, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 960 (2020) 115193 [1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00392].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Vladimirov, Correspondence between Soft and Rapidity Anomalous Dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 118 (2017) 062001 [1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='05791].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [29] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Li and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zhu, Bootstrapping Rapidity Anomalous Dimensions for Transverse-Momentum Resummation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 118 (2017) 022004 [1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='01404].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [30] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Billis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ebert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Michel and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Tackmann, A toolbox for qT and 0-jettiness subtractions at N3LO, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Plus 136 (2021) 214 [1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00811].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [31] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neumann, Fiducial qT resummation of color-singlet processes at N3LL+NNLO, JHEP 03 (2021) 199 [2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11437].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Chiu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Jain, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neill and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rothstein, A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory, JHEP 05 (2012) 084 [1202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0814].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Chiu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Jain, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neill and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rothstein, The Rapidity Renormalization Group, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 108 (2012) 151601 [1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='0881].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Alioli and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Walsh, Jet Veto Clustering Logarithms Beyond Leading Order, JHEP 03 (2014) 119 [1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5234].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Dasgupta, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Dreyer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Salam and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Soyez, Small-radius jets to all orders in QCD, JHEP 04 (2015) 039 [1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='5182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [36] NNPDF collaboration, Parton distributions from high-precision collider data, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C 77 (2017) 663 [1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00428].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Banfi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Monni and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zanderighi, Quark masses in Higgs production with a jet veto, JHEP 01 (2014) 097 [1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4634].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 55 – [38] CMS collaboration, Measurement of differential cross sections for the production of a Z boson in association with jets in proton-proton collisions at √s = 13 TeV, 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='02872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [39] CMS collaboration, W+W− boson pair production in proton-proton collisions at √s = 13 TeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' D 102 (2020) 092001 [2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='00119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [40] CMS collaboration, Measurements of pp → ZZ production cross sections and constraints on anomalous triple gauge couplings at √s = 13 TeV, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C 81 (2021) 200 [2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='01186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [41] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Jaiswal and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Okui, Reemergence of rapidity-scale uncertainty in soft-collinear effective theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' D 92 (2015) 074035 [1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='07529].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Michel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Pietrulewicz and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Tackmann, Jet Veto Resummation with Jet Rapidity Cuts, JHEP 04 (2019) 142 [1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='12911].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [43] ATLAS collaboration, Measurement of differential cross sections and W +/W − cross-section ratios for W boson production in association with jets at √s = 8 TeV with the ATLAS detector, JHEP 05 (2018) 077 [1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='03296].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [44] ATLAS collaboration, Measurement of W ±Z production cross sections and gauge boson polarisation in pp collisions at √s = 13 TeV with the ATLAS detector, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C 79 (2019) 535 [1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='05759].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [45] CMS collaboration, Measurement of the inclusive and differential WZ production cross sections, polarization angles, and triple gauge couplings in pp collisions at √s = 13 TeV, JHEP 07 (2022) 032 [2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='11231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Banfi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Salam and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zanderighi, Principles of general final-state resummation and automated implementation, JHEP 03 (2005) 073 [hep-ph/0407286].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [47] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Monni, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rottoli and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Torrielli, Higgs transverse momentum with a jet veto: a double-differential resummation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 124 (2020) 252001 [1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='04704].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [48] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Bizon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Monni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Re, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rottoli and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Torrielli, Momentum-space resummation for transverse observables and the Higgs p⊥ at N3LL+NNLO, JHEP 02 (2018) 108 [1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='09127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [49] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Monni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Re and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Torrielli, Higgs Transverse-Momentum Resummation in Direct Space, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 116 (2016) 242001 [1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='02191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [50] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neubert and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Wilhelm, Electroweak Gauge-Boson Production at Small qT : Infrared Safety from the Collinear Anomaly, JHEP 02 (2012) 124 [1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6027].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [51] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ahrens, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neubert and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Yang, Origin of the Large Perturbative Corrections to Higgs Production at Hadron Colliders, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' D 79 (2009) 033013 [0808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [52] ATLAS collaboration, Measurement of the W +W − production cross section in pp collisions at a centre-of -mass energy of √s = 13 TeV with the ATLAS experiment, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 773 (2017) 354 [1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='04519].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [53] ATLAS collaboration, Measurement of fiducial and differential W +W − production cross-sections at √s = 13 TeV with the ATLAS detector, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C 79 (2019) 884 [1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='04242].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [54] CMS collaboration, Search for anomalous triple gauge couplings in WW and WZ production in lepton + jet ev ents in proton-proton collisions at √s = 13 TeV, JHEP 12 (2019) 062 [1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='08354].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [55] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Campbell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ellis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neumann and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Seth, Transverse momentum resummation at N3LL+NNLO for diboson processes, 2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [56] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Bonciani, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Del Duca, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Frellesvig, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Hidding, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Hirschi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Moriello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=', – 56 – Next-to-leading-order QCD Corrections to Higgs Production in association with a Jet, 2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [57] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Czakon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Harlander, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Klappert and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Niggetiedt, Exact Top-Quark Mass Dependence in Hadronic Higgs Production, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 127 (2021) 162002 [2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='04436].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [58] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neumann and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Wiesemann, Finite top-mass effects in gluon-induced Higgs production with a jet-veto at NNLO, JHEP 11 (2014) 150 [1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='6836].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [59] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Idilbi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ji and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Yuan, Transverse momentum distribution through soft-gluon resummation in effective field theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 625 (2005) 253 [hep-ph/0507196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [60] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Idilbi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ji, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ma and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Yuan, Threshold resummation for Higgs production in effective field theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' D 73 (2006) 077501 [hep-ph/0509294].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [61] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ahrens, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neubert and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Yang, Renormalization-Group Improved Prediction for Higgs Production at Hadron Colliders, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C 62 (2009) 333 [0809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4283].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [62] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Mantry and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Petriello, Factorization and Resummation of Higgs Boson Differential Distributions in Soft-Collinear Effective Theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' D 81 (2010) 093007 [0911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [63] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Bell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Brune, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Das and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Wald, The NNLO quark beam function for jet-veto resummation, 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='05578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [64] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Luo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Xu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Yang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Yang and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zhu, Transverse Parton Distribution and Fragmentation Functions at NNLO: the Quark Case, JHEP 10 (2019) 083 [1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='03831].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [65] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Luo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zhu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zhu, Transverse Parton Distribution and Fragmentation Functions at NNLO: the Gluon Case, JHEP 01 (2020) 040 [1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='13820].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [66] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neubert, Drell-Yan Production at Small qT , Transverse Parton Distributions and the Collinear Anomaly, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C 71 (2011) 1665 [1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='4005].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [67] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Tarasov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Vladimirov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zharkov, The Gell-Mann-Low Function of QCD in the Three Loop Approximation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 93 (1980) 429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [68] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Larin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Vermaseren, The Three loop QCD Beta function and anomalous dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 303 (1993) 334 [hep-ph/9302208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [69] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' van Ritbergen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Vermaseren and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Larin, The Four loop beta function in quantum chromodynamics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 400 (1997) 379 [hep-ph/9701390].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [70] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' van Ritbergen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Schellekens and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Vermaseren, Group theory factors for Feynman diagrams, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' A 14 (1999) 41 [hep-ph/9802376].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [71] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Henn, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Korchemsky and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Mistlberger, The full four-loop cusp anomalous dimension in N = 4 super Yang-Mills and QCD, JHEP 04 (2020) 018 [1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [72] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' von Manteuffel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Panzer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Schabinger, Cusp and collinear anomalous dimensions in four-loop QCD from form factors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 124 (2020) 162001 [2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='04617].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [73] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Broggio and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ferroglia, Introduction to Soft-Collinear Effective Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 896, Springer (2015), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1007/978-3-319-14848-9, [1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1892].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [74] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Becher and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Neubert, On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP 06 (2009) 081 [0903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [75] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Altarelli and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Parisi, Asymptotic Freedom in Parton Language, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 126 (1977) 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 57 – [76] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Curci, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Furmanski and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Petronzio, Evolution of Parton Densities Beyond Leading Order: The Nonsinglet Case, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 175 (1980) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [77] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Furmanski and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Petronzio, Singlet Parton Densities Beyond Leading Order, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 97 (1980) 437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [78] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ellis, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Stirling and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Webber, QCD and collider physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 8, Cambridge University Press (2, 2011), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='1017/CBO9780511628788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [79] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Kramer and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lampe, Two Jet Cross-Section in e+ e- Annihilation, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C 34 (1987) 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [80] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Matsuura and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' van Neerven, Second Order Logarithmic Corrections to the Drell-Yan Cross-section, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' C 38 (1988) 623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [81] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Matsuura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' van der Marck and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' van Neerven, The Calculation of the Second Order Soft and Virtual Contributions to the Drell-Yan Cross-Section, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 319 (1989) 570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [82] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Moch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Vermaseren and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Vogt, Three-loop results for quark and gluon form-factors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 625 (2005) 245 [hep-ph/0508055].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [83] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Moch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Vermaseren and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Vogt, The Quark form-factor at higher orders, JHEP 08 (2005) 049 [hep-ph/0507039].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [84] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Gehrmann, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Glover, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Huber, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ikizlerli and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Studerus, Calculation of the quark and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='3653].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [85] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Idilbi and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ji, Threshold resummation for Drell-Yan process in soft-collinear effective theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' D 72 (2005) 054016 [hep-ph/0501006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [86] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Idilbi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ji and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Yuan, Resummation of threshold logarithms in effective field theory for DIS, Drell-Yan and Higgs production, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 753 (2006) 42 [hep-ph/0605068].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [87] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Campbell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ellis and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Seth, Non-local slicing approaches for NNLO QCD in MCFM, JHEP 06 (2022) 002 [2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='07738].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [88] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Davies, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Herren and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Steinhauser, Top Quark Mass Effects in Higgs Boson Production at Four-Loop Order: Virtual Corrections, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' 124 (2020) 112002 [1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='10214].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [89] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Spira, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Djouadi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Graudenz and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Zerwas, Higgs boson production at the LHC, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 453 (1995) 17 [hep-ph/9504378].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [90] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Harlander and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Kant, Higgs production and decay: Analytic results at next-to-leading order QCD, JHEP 12 (2005) 015 [hep-ph/0509189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [91] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Anastasiou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Beerli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Bucherer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Daleo and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Kunszt, Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop, JHEP 01 (2007) 082 [hep-ph/0611236].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [92] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Harlander and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Ozeren, Top mass effects in Higgs production at next-to-next-to-leading order QCD: Virtual corrections, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 679 (2009) 467 [0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2997].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' [93] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Pak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Rogal and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Steinhauser, Virtual three-loop corrections to Higgs boson production in gluon fusion for finite top quark mass, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' B 679 (2009) 473 [0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content='2998].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} +page_content=' – 58 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQfQy0a/content/2301.11768v1.pdf'} diff --git a/4NE1T4oBgHgl3EQf6AXQ/content/tmp_files/2301.03519v1.pdf.txt b/4NE1T4oBgHgl3EQf6AXQ/content/tmp_files/2301.03519v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec357fecee664c3ac70405c0d19ba999c798e57a --- /dev/null +++ b/4NE1T4oBgHgl3EQf6AXQ/content/tmp_files/2301.03519v1.pdf.txt @@ -0,0 +1,1374 @@ +Draft version January 10, 2023 +Typeset using LATEX twocolumn style in AASTeX63 +Evolution of elemental abundances in hot active region cores from Chandrayaan-2 XSM observations +Biswajit Mondal,1, 2 Santosh V. Vadawale,1 Giulio Del Zanna,3 N. P. S. Mithun,1 Aveek Sarkar,1 +Helen E. Mason,3 P. Janardhan,1 and Anil Bhardwaj1 +1Physical Research Laboratory, Navrangpura, Ahmedabad, Gujarat-380 009, India +2Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat-382 355, India +3DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK +ABSTRACT +The First Ionization Potential (FIP) bias, whereby elemental abundances for low FIP elements in +different coronal structures vary from their photospheric values and may also vary with time, has been +widely studied. In order to study the temporal variation, and to understand the physical mechanisms +giving rise to the FIP bias, we have investigated the hot cores of three ARs using disk-integrated +soft X-ray spectroscopic observation with the Solar X-ray Monitor (XSM) onboard Chandrayaan-2. +Observations for periods when only one AR was present on the solar disk were used so as to ensure that +the AR was the principal contributor to the total X-ray intensity. The average values of temperature +and EM were ∼3 MK and 3×1046 cm−3 respectively. Regardless of the age and activity of the AR, +the elemental abundances of the low FIP elements, Al, Mg, and Si were consistently higher than their +photospheric values. The average FIP bias for Mg and Si was ∼3, whereas the FIP bias for the mid-FIP +element, S, was ∼1.5. However, the FIP bias for the lowest FIP element, Al, was observed to be higher +than 3, which, if real, suggests a dependence of the FIP bias of low FIP elements on their FIP value. +Another major result from our analysis is that the FIP bias of these elements is established in within +∼10 hours of emergence of the AR and then remains almost constant throughout its lifetime. +Keywords: Solar X-ray corona, Solar abundances, FIP bias, FIP effect, Active Region +1. INTRODUCTION +The earlier study of the Sun as a star by Pottasch +(1963) revealed that solar coronal abundances are dif- +ferent from those of the photosphere. The differences +are correlated to the First Ionization Potential (FIP) of +the element, in the sense that the abundance ratio of +a low-FIP (less than 10 eV) element versus that of a +high-FIP element is higher in the corona. A measure +of the difference is the so called FIP bias, i.e. the ratio +between the coronal and the photospheric abundance of +an element. +In most of the available literature, the FIP bias has +been (and still is) estimated by measuring the relative +abundances between elements, and not relative to hy- +drogen. This is due to the fact that abundance mea- +surements with respect to Hydrogen in the low corona, +and on-disk is non-trivial, due to the lack of H-emission +Corresponding author: Biswajit Mondal +biswajit70mondal94@gmail.com, biswajitm@prl.res.in +lines at a few million Kelvin. Hence, whether it is the +low-FIP elements that have an increased abundance or +the high-FIP elements that have a reduced one (com- +pared to their photospheric values) has been a subject +of continued debate. +Further, it has become clear that different solar struc- +tures have different FIP biases. There are also indica- +tions that the FIP bias depends on the temperature of +the plasma. For a long time, it has been widely accepted +that coronal abundances in active regions increase with +time. We refer the reader to the recent reviews by Lam- +ing (2015); Del Zanna and Mason (2018) for more de- +tails. We also provide in the following section a brief +summary of available measurements related to active re- +gions. +Knowledge of the elemental abundances in different +atmospheric layers of the Sun is a topic of great inter- +est to the solar physics community mainly due to the +following two reasons. The first is that they provide, in +principle, a way to link the solar source regions to the +various components of the solar wind. In fact, elemental +abundance variations are also clearly observed in-situ. +arXiv:2301.03519v1 [astro-ph.SR] 9 Jan 2023 + +2 +The slow-speed solar wind has a high FIP bias simi- +lar to that measured in AR core loops, 3MK, whereas +the high-speed wind has a near unit FIP bias, similar +to that of coronal holes (see, e.g., Brooks et al. 2015; +Gloeckler and Geiss 1989; Feldman et al. 1998; Bochsler +2007; Brooks and Warren 2011). +The second reason is that studying abundance vari- +ations might contribute to a better understanding of +the physical processes at play in the solar corona. In +fact, we know that the FIP bias is closely related to +the magnetic field activity of the Sun (see, e.g. Feld- +man and Widing 2002; Brooks et al. 2017; Baker et al. +2018). The Ponderomotive force model (Laming 2004, +2009, 2012, 2017) is now widely accepted, as it is able to +reproduce the main characteristics of the FIP effect, as +measured in-situ and remotely. According to this model, +the separation of ions from neutral atoms within closed +loops in an upward direction is caused by the reflec- +tion of downward propagating Alfv’en waves at chromo- +spheric heights, causing an enhancement of the low-FIP +elements in the corona. Since coronal waves can be pro- +duced by mechanisms that heat the solar corona, it is +thought that the mechanism underlying the FIP effect +is inextricably linked to processes that heat the solar +corona. +Hence, measuring the FIP bias is an impor- +tant diagnostic for coronal plasma characteristics (Lam- +ing 2015; Dahlburg et al. 2016). +In this paper, we focus on the elemental abundances +of hot, quiescent AR core emission at 3 MK, by provid- +ing line-to-continuum measurements of the Sun in the +soft X-ray energy band using data from the Solar X-ray +Monitor (XSM: Vadawale et al. 2014; Shanmugam et al. +2020). It may be noted here that the XSM is the only +spectrometer to have observed the Sun in the 1-15 keV +range during the minimum of solar cycle 24 with an en- +ergy resolution better than 180 eV at 5.9 keV. This reso- +lution is sufficient to measure the abundances of several +elements. The soft X-ray continuum is dominated by +free-free radiation (with some free-bound emission, see +e.g. Figure 12b of Mondal et al. 2021), which primarily +originates from H. Hence, measuring the abundances of +an emission line with respect to the continuum provides +the absolute abundance of that element. It should be +noted that the measurement of free-free emission can +also be carried out in the EUV energy band, but it is +limited to large flares (e.g., Feldman et al. 2003). +The XSM energy band is sensitive to temperatures +above 2 MK. When the Sun was at minimum activity +levels, without any ARs, the XSM observed a steady sig- +nal originating from X-ray Bright Points (XBPs), with +a peak emission around 2 MK (Vadawale et al. 2021b). +When a single non-flaring AR is present, the signal is +dominated by the AR’s near-isothermal ∼ 3 MK emis- +sion (see, e.g. Del Zanna 2013). This provides an ex- +cellent opportunity to measure the FIP bias of the hot +quiescent core for individual active regions during their +evolution. +In the literature, very few abundance measurements +are know to be associated specifically with the 3 MK +emission from quiescent AR cores. +These are sum- +marised in Del Zanna and Mason (2018). X-ray spectra +in the 10–20 ˚A range have provided the relative abun- +dances of the low-FIP Fe, Mg vs. O, Ne. Most stud- +ies provided results on single active regions. Saba and +Strong (1993) reported a significant variability of the +FIP bias using SMM/FCS observations of several active +regions. On the other hand, a re-analysis of several qui- +escent AR cores with improved atomic data and using +a multi-thermal DEM technique by Del Zanna and Ma- +son (2014) indicated the same FIP bias, around 3, for +all active regions, irrespective of their age and size. +Since 2006, EUV spectra from the Hinode EIS instru- +ment have provided an opportunity to measure the rel- +ative FIP bias between low-FIP elements (e.g. Fe, Si) +and the high-FIP Ar, as well as the mid-FIP S, which +actually shows the same abundance variations as the +high-FIP elements. An example case was discussed by +Del Zanna (2013), showing that the FIP bias in the EUV +of 3 MK plasma was the same as in the X-rays. Con- +sidering the size of the emitting plasma and its emission +measure, Del Zanna (2013) concluded that it should be +the low-FIP elements that are over-abundant by about +a factor of 3. +Del Zanna et al. (2022) carried out a multi-wavelength +study of an AR as it crossed the solar disk which was ob- +served by XSM as well as by SDO/AIA, Hinode/EIS and +Hinode/XRT. The relative FIP bias obtained from Hin- +ode/EIS observations confirmed the Del Zanna (2013) +results, and showed no variation with the disk passage. +The analysis of simultaneous XSM spectra on two days +also indicated no significant variability, and provided an +absolute FIP bias for Si of 2.4, i.e. close to the value +suggested by Del Zanna (2013), and also very close to +the prediction of Laming’s model. +In the present study, we extend the previous XSM +analysis to all the quiescent periods of the same active +region, and also investigate two other active regions dur- +ing their disk crossings. One AR in particular is of in- +terest as it emerged on-disk, and hence offers the op- +portunity to study the elemental abundances during the +early phase of the evolution of an AR. +The rest of the paper is organized as follows: Sec- +tion 2 provides a short overview of previous abundance +measurements in active regions. Section 3 describes the + +3 +observations and data analysis. +Section 4 provides a +detailed spectral analysis. After obtaining the results, +these are discussed in Section 5. Section 6 provides a +brief summary of the article. +2. HISTORICAL OVERVIEW +Spatially resolved measurements of the relative FIP +bias have been carried out by several authors (see,e.g. +Widing and Feldman 1993; Sheeley 1995, 1996; Widing +1997; Widing and Feldman 2001) using Skylab spectro- +heliograms with Mg, Ne transition region lines. These +are formed well below 1 MK, in the legs of active re- +gion ‘cool’ (1 MK) loops. They found photospheric com- +position (FIP bias=1) for newly emerged closed loops, +but increasing FIP bias reaching a value of 3-4 within a +timescale of 1-2 days (Widing and Feldman 2001), and +much higher values, up to about 10, within a few more +days. Differing FIP biases were also obtained by Young +and Mason (1997) and Dwivedi et al. (1999) using Mg +and Ne line ratios observed by the CDS and SUMER +spectrometers onboard the Solar and Heliospheric Ob- +servatory (SOHO). +The large values are hard to reconcile with in-situ +measurements, where the FIP bias is at most 3, and +also with theory. However, Del Zanna (2003) pointed +out that as the cool AR loops are almost isothermal in +their cross-section, the assumption that a smooth emis- +sion measure distribution was present in the plasma, +used to interpret the Skylab data, was not justified. +Del Zanna (2003) took the intensities measured by Wid- +ing and Feldman (1993), and using an emission measure +loci approach, showed that a FIP bias of 3.7 was con- +sistent with the data, much lower than the value of 14 +reported by Widing and Feldman. +Del Zanna (2003) +also analysed the legs of several cool loops observed +by SoHO/CDS and found photospheric abundances, al- +though a similar analysis for other loops by Del Zanna +and Mason (2003) found a FIP bias of 4. +In summary, the legs of cool AR loops do show a range +of FIP bias values, between 1 and 4, and perhaps occa- +sionally larger. However, the very high FIP biases found +from Skylab data were largely overestimated. +As shown by Del Zanna and Mason (2003), active re- +gion cores are composed not only of cool 1 MK loops +and unresolved, almost isothermal 3 MK loops, but also +unresolved emission in the 1–3 MK range. The plasma +at different temperatures is generally not cospatial. +There is evidence from Hinode EIS observations of e.g. +Si X, S X lines that this ≃2 MK emission has a lower +relative FIP bias, around 2 (see,e.g. Del Zanna 2012). +Further studies using the same lines (e.g., Baker et al. +2013, 2015; Doschek and Warren 2019; Mihailescu et al. +2022; Ko et al. 2016; Testa et al. 2022) have shown some +variation (around the value of 2) of the relative FIP bias +within each active region, but little variability in time, +except during the decay phase, when an AR effectively +disappears and the relative abundances become photo- +spheric. +In summary, active region structures formed at tem- +peratures below 2 MK show a range of relative FIP bi- +ases, and some temporal variability. The few observa- +tions of the hotter, 3 MK, AR cores have in contrast +shown a remarkable consistency, with relative FIP bi- +ases around 3. +Finally, to interpret observations of the Sun as a star, +one needs to take into account the above (and other) +issues. As shown by Del Zanna (2019), when the Sun’s +actvity is at a minimum with no active region present +on the solar disk, the corona around 1 MK shows near +photospheric abundances, whereas in presence of active +regions, the FIP bias for the 1 MK emission stays the +same, but the hotter emission shows a higher relative +FIP bias. +When active regions flare, the high tem- +perature plasma shows nearly photospheric composition +around the peak X-ray emission (see e.g., Mondal et al. +2021). +3. OBSERVATIONS AND DATA ANALYSIS +Observations of the Sun were carried out with the +XSM during the minimum of solar cycle 24, when no +active regions were present, covering the years 2019- +2020. Results are given in Vadawale et al. (2021b). They +reported intermediate abundances of low-FIP elements +(Mg, Al, and Si) of 2 MK plasma, primarily originating +from X-ray Bright Points, XBPs (Mondal et al. 2022). +Frequent micro-flaring activity was observed and found +to be occurring everywhere on the solar disk, even when +no ARs were present (Vadawale et al. 2021a). During +the minimum of solar cycle 24, XSM observed the disk +passage of a few individual, isolated ARs in the absence +of any other major activity. When ARs were present +on-disk, XSM recorded hundreds of small flares of dif- +ferent classes. Elemental abundance variations during +these small flares were found, for the first time, to ini- +tially drop to photospheric values, then rapidly return +to coronal values, as described by Mondal et al. (2021), +Mithun et al. (2022), and Lakshitha et al. (2022). In +this paper, we analyze the temporal evolution of active +regions outside of flaring activity and for this we have +chosen to study three isolated active regions: AR12749, +AR12758, and AR12759. +XSM data contain spectra at 1 s cadence in a raw +(level-1) daily file. Since the visibility of the Sun varies +within the XSM field-of-view (FOV), with the Sun be- + +4 +ing sometimes outside the FOV or being occulted by the +Moon, the data include both solar and non-solar spectra. +The XSM Data Analysis Software (XSMDAS: Mithun +et al. (2021)) has been used to generate the level-2 sci- +ence data product using the appropriate Good Time +Intervals (GTIs) and the other necessary instrumental +parameters. The available default level-2 data contains +the effective area corrected light curves for every second +and spectra for every minute. XSMDAS also provides +the functionality to generate the light curves and spec- +tra for a given cadence and energy range, which we have +used in the present analysis. +Using the XSMDAS, we have generated 2 min av- +eraged XSM light curves in the energy range of 1-15 +keV during the disk passage of the AR12749, AR12758, +and AR12759, as shown in the three panels of Figure 1. +During the evolution of these three ARs, representative +full disk X-ray images taken by the XRT Be-thin fil- +ter are shown in the top row of each panel. AR12749 +(Figure 1a) appeared from the east limb on Sept 29, +2019. Whilst crossing the solar disk, it became fainter +towards the west limb and went behind the limb on 14 +Oct. AR12758 (Figure 1b) appears to form on disk on +06 Mar 2020 and fully emerged after 08 Mar. It decays +whilst crossing the solar disk and finally goes behind the +west limb on 18 Mar. AR12759 appeared from the east +limb on 29 Mar 2020 and transited the solar disk until +14 Apr 2020, before disappearing behind the west limb. +The full disk XRT images show that during the pas- +sage of these three ARs, no other major activity was +present on the solar disk. Thus, we conclude that these +three ARs were primarily responsible, during their disk +passage, for the enhanced X-ray emission observed by +the XSM. These ARs produced many small B/A-class +flares, seen as multiple spikes in the XSM light curves. +Detailed studies of these small flares were reported by +Mondal et al. (2021) and Lakshitha et al. (2022). +In the present study, we have selected only the quies- +cent periods from the observed light curves by exclud- +ing the periods when the small flares occurred using a +semi-automated graphical algorithm. For example, Fig- +ure 2 shows the representative selection (orange shaded +regions) for the AR quiescent durations on 2020-04-06. +These identified time intervals were used as user-defined +GTIs to generate the spectra for quiescent ARs on a +daily basis in order to carry out the detailed spectral +analysis as discussed in Section 4. +4. SPECTRAL ANALYSIS +Broad-band soft X-ray spectra of the solar corona con- +sist of a continuum as well as the emission lines of the +different elements. Modeling the soft X-ray spectrum +provides the measurements of the temperature, emission +measure, and elemental abundances (with respect to hy- +drogen) of the emitting plasma (Del Zanna and Mason +2018). We use the chisoth model (Mondal et al. 2021) +for the spectral fitting. +The chisoth is a local model +of the X-ray spectral fitting package (XSPEC: Arnaud +et al. (1999)), and it estimates the theoretical spectrum +using the CHIANTI atomic database. It takes temper- +ature, emission measure (EM: which is related to the +density of the plasma), and the elemental abundances +of the elements from Z=2 to Z=30 as free variables for +the spectral fitting. +After generating the spectra for the quiescent peri- +ods, we fitted them with an isothermal emission model. +For the spectral fitting, we ignored the spectra below 1.3 +keV where the XSM response is not well-known (Mithun +et al. 2020), and above the energy where the solar spec- +trum is dominated by the non-solar background spec- +trum. During the spectral fitting, the temperature, EM, +along with the abundances of Mg, Al, and Si (whose +emission lines are prominent in the XSM spectrum) were +kept as variable parameters. The 1σ uncertainty of each +free parameter was also estimated using the standard +procedure in XSPEC. +Although the S line complex is visible in the spectra, +including it in the spectral fits as a free parameter causes +a large uncertainty in the measurement of the S abun- +dance because of its poor statistics. +Hence, we fixed +the S abundances along with the abundances of other +elements (whose emission lines are not visible in the ob- +served spectra) with the coronal abundances of Feldman +(1992). However, we found that the measurement of the +S abundance is possible for the summed spectrum of the +entire AR period. +Figure 3 shows the representative XSM spectra, for +the three ARs fitted, in different colours, with an isother- +mal model. +The points with error bars represent the +observed spectra, whereas the solid curves represent the +best-fit modeled spectra. The grey error bars represent +the non-solar background spectrum, which is subtracted +from the observed spectra during the spectral analysis. +The lower panel shows the residual between the observed +and model spectra. We have fitted all the spectra in a +similar way and found that all of them are well described +by isothermal model. +The X-rays observed by XSM originated from both +the AR and the background quiet Sun regions (outside +the AR). To determine how much emission is due to the +background quiet Sun regions, we estimate the average +quiet Sun spectrum using an average quiet-Sun temper- +ature, EM, and abundances, as reported by Vadawale +et al. (2021b). The average quiet Sun spectrum is shown + +5 +Sep-29 +Oct-01 +Oct-03 +Oct-05 +Oct-07 +Oct-09 +Oct-11 +Date (2019) +101 +102 +103 +XSM Counts (s +1) +AR12749 +a +Mar-06 +Mar-08 +Mar-10 +Mar-12 +Mar-14 +Mar-16 +Mar-18 +Date (2020) +101 +102 +103 +XSM Counts (s +1) +AR12758 +b +Mar-26 +Mar-28 +Mar-30 +Apr-01 +Apr-03 +Apr-05 +Apr-07 +Apr-09 +Apr-11 +Apr-13 +Date (2020) +101 +102 +103 +XSM Counts (s +1) +AR12759 +c +Figure 1. XSM 1-15 keV light curves during the disk passage of AR12749 (panel a), AR12758 (panel b) and AR12759 (panel +c). The top row of each panel shows representative full disk X-ray images (negative intensities) taken with the XRT Be-thin +filter during the evolution of the ARs. The vertical dashed lines represent the timing of the XRT images. + +6 +05:33 +11:06 +16:40 +22:13 +hh:mm on 2020-04-05 +100 +101 +102 +103 +Rate(c/s) +c +05:33 +11:06 +16:40 +22:13 +hh:mm on 2020-03-11 +100 +101 +102 +103 +Rate(c/s) +b +05:33 +11:06 +16:40 +22:13 +hh:mm on 2019-10-01 +100 +101 +102 +103 +Rate(c/s) +a +Figure 2. Selection of the quiescent AR periods (orange- +shaded regions) from the XSM light-curves for one represen- +tative day of AR12749 (panel a), AR12758 (panel b), and +AR12759 (panel c). +by the black dashed curve in Figure 3. The quiet Sun +spectrum is found to be almost an order of magnitude +lower than the spectra of the active period when the +ARs were very bright on the solar disk. We thus con- +clude that the X-ray emission of the active periods is +primarily dominated by the AR emission. +Separating the AR emission from the background +quiet Sun emission would be possible by subtracting the +quiet-sun spectra from the AR spectra. But, as the ef- +fective area of the XSM varies with time, this is not +recommended. It is possible to model the AR spectra +using a two-temperature (2T) component model rather +than subtracting the quiet Sun spectra. This is what we +have chosen to do. One temperature corresponds to the +background solar emission originating from the regions +outside the AR and the second temperature corresponds +to the AR plasma. We have modeled a few AR spec- +tra with a two-temperature (2T) model. During the 2T +spectral fitting, the parameters of the background solar +emission were kept fixed to the average quiet-Sun values +reported by Vadawale et al. (2021b). For the AR compo- +nent, the temperature, EM, along with the abundances +of Mg, Al, and Si, were kept as variable parameters. We +found that the 2T model can describe the XSM spectra +for the active periods with similar best-fitted parameters +as those obtained by the isothermal model. This verifies +that the AR emission dominates the spectra of the AR +periods. Thus, in this study, we show the results of the +isothermal analysis in Figure 5 and 6. This is discussed +in Section 5. +It is interesting to study how the plasma parameters +vary during the emerging phase of the AR12758, i.e., +from 07-Mar-2020 to 09-Mar-2020. Figure 4 shows the +evolution of the photospheric magnetograms (top row) +and the X-ray emission (bottom row) as observed by +SDO/HMI and the Be-thin filter of Hinode/XRT re- +spectively. +These images were created by de-rotating +the synoptic data of HMI1 and XRT2 to a common date +(08-Mar-2020) using the standard procedure of Solar- +SoftWare (SSW; Freeland and Handy 1998). We also +determined the total unsigned photospheric magnetic +flux for the regions ±10 G within the field-of-view shown +in Figure 4. During this emerging flux period, we car- +ried out a time-resolved spectroscopic study using the +XSM observations with finer time bins of less than a +day. However, during this period, as the emission from +the AR was not very bright, the emission from the AR +and the rest of the Sun could be mixed together. Thus +to derive the evolution of the plasma parameters during +this period, we modeled the observed XSM spectra with +a 2T model, where one component represents the emis- +sion from the AR, and the other represents the emission +from the rest of the Sun, as discussed in the previous +paragraph. The results are shown in Figure 7 and dis- +cussed in Section 5. +5. RESULTS AND DISCUSSION +In this study, we have performed the X-ray spectral +analysis for the evolution of three ARs as observed by +the XSM. The AR spectra (Figure 3) show a clear sig- +nature of the thermal X-ray emission from the line com- +1 http://jsoc.stanford.edu/data/hmi/synoptic/ +2 http://solar.physics.montana.edu/HINODE/XRT/SCIA/ + +7 +10 +2 +10 +1 +100 +101 +Counts (s +1keV +1) +Mg +Mg / Al +Si +Si +S +Quiet Sun +AR12749 (Oct-01) +AR12758 (Mar-11) +AR12759 (Apr-05) +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +3.00 +3.25 +Energy (keV) +5 +0 +5 +Figure 3. Soft X-ray spectra measured by the XSM for three +representative days of the AR period are shown. Solid lines +represent the best-fit isothermal model, and the residuals are +shown in the bottom panel. Gray points correspond to the +non-solar background spectrum. +plexes of Mg, Al, Si, and S, along with the continuum +emission up to ∼3.0 keV. The red points in Figure 5 +show the evolution of the temperature and EM through- +out the evolution of the three ARs. Figure 6 shows the +evolution of abundances of Mg (panel a), Al (panel b), +and Si (panel c). The error bars associated with all the +parameters along the y-axis represent the 1σ uncertain- +ties. +We also derived the average S abundance along +with the other elements from the summed spectrum for +the duration when the ARs were very bright on the solar +disk (bounded by the vertical dashed lines in Figures 5 +and 6). +This provides the average parameters associ- +ated with each AR, as shown by magenta bars and also +given in Table 1. The primary findings of the paper are +discussed below. +5.1. Temperature and emission measure +Temperatures (T) and emission measures (EM) are +close to the quiet Sun levels (black dashed lines in Fig- +ure 5) when the ARs were absent from the solar disc +or only partially present, e.g., 30 September 2019 and +6 March 2020. Once the ARs appear, the temperature +rises to more than ∼3 MK from the ∼2 MK of the quiet +Sun. As the ∼3 MK emission is predominantly derived +from a smaller volume of AR plasma, the presence of +the AR reduces the EM from the quiet Sun values. The +average temperatures for all the ARs are determined to +be ∼3 MK (blue error bars in Figure 5a), which is close +to the “basal” temperature of the AR core reported in +earlier research (e.g., Del Zanna and Mason 2018; Del +Zanna 2012; Winebarger et al. 2012). The temperature +and EM do, however, vary slightly over the course of the +AR’s evolution, which is consistent with the observed X- +ray light curve. Following the arrival of AR12749 and +AR12758, their activity decayed while rotating on the +solar disk (Figure 1), which is why the temperature and +EM decreased during their evolution, as indicated by the +dashed vertical lines in Figure 5. After October 6, 2019, +the EM for AR12749 begins to rise as the AR weakens +and the quiet Sun emission takes precedence over the +AR emission. Thus, after the AR has almost died and is +very faint, the EM and temperature reach values close +to the quiet Sun temperature and EM. The temperature +and EM for the AR12759 remain almost constant with +time, as this AR crossed the solar disk without much +decay in activity (Figure 1c). +5.2. Abundance evolution +In contrast to the temperature and EM, the abun- +dances of Mg, Al, and Si do not follow the X-ray light +curve of any of the three ARs throughout their evolu- +tion (Figure 6). The abundances obtained for low-FIP +elements Al, Mg, and Si are consistently greater than +the photospheric values, demonstrating a persistent FIP +bias during the course of the AR. After the emergence +of AR12758, the FIP bias is found to be almost constant +throughout its decay phase. Similarly, during the decay +of the AR12749, the FIP bias remains nearly constant, +in contrast to certain earlier studies, such as Ko et al. +(2016). +They suggested decreasing FIP bias in high- +temperature plasma of more than two million degrees +during the decay phase of an AR. The more established +AR, AR12759, which evolved without decaying much +during its transit across the solar disk, also shows an +almost constant FIP bias, similar to the other two ARs. +We do not find any relationship between the age of +the AR and the FIP bias, as suggested in some previous +papers, e.g.,Del Zanna and Mason 2014; Doschek and +Warren 2019. +The measured abundances for Mg, Si, +and S are comparable to those given by (Feldman 1992) +and Fludra and Schmelz (1999) (orange shaded regions +in Figure 6). However, the Al abundance is ∼30%-60% +higher than the coronal abundances reported in the lit- +erature. We note that the Al lines in the XSM spectra +are blended with Mg lines. From Markov Chain Monte +Carlo (MCMC) analysis (discussed in Appendix A), we +find that there is no anti-correlation between Mg and +Al abundances. This suggests that the observed spectra +does indeed require higher abundances of Al and cannot +be explained by an enhancement of Mg abundances. +5.3. FIP bias at the onset of AR core +Though we do not find any relationship between the +age of the AR cores and their FIP biases (Section 5.2), + +8 + 5-Mar 17:58 + 6-Mar 05:58 + 7-Mar 02:58 + 7-Mar 12:58 + 7-Mar 06:58 + 8-Mar 01:58 +Figure 4. Evolution of the AR12758 during its emergence phase on the solar disk. Top row shows the evolution of photospheric +magnetograms as observed by HMI and bottom row shows the evolution of X-ray emission as observed by XRT Be-thin filter. +Figure 5. +Evolution of the temperature (red points in panel a) and EM (red points in panel b) during the evolution of +AR12749, AR12758, and AR12759. When the ARs were very bright, as bounded by the vertical dashed lines, the magenta bars +represent the average values of the temperature and EM. The black horizontal dashed lines represent the average temperature +and emission measure for the quiet Sun in the absence of any AR reported by Vadawale et al. (2021b). The XSM lightcurves +of the ARs are shown in grey color, and the lightcurves for the quiescent regions are shown in blue colors. +which remain constant, it is interesting to study the +timescale on which the FIP bias developed during the +emergence of the AR core. Such a study has been made +possible using the finer (< one day) time-resolved spec- +troscopy during the emerging phase (07-Mar-2020 to 09- +Mar-2020) of AR12758. +During this period, we esti- +mated the total unsigned photospheric magnetic flux as +measured by HMI/SDO and shown in Figure 7a (black +color). +The peak in the magnetic flux represents the +time when the AR completely emerged into the solar +disk. +After the emergence, the unsigned magnetic flux is +found to (temporarily) decrease. Figures 7b and 7c show +the evolution of the AR core temperature and emission- +measure. With the emergence of the AR. The temper- +ature becomes close to the AR core temperature of ∼3 + +AR12749 +AR12758 +AR12759 +4.0 +a +3.5 +(MK) +3.0 +2.5 +2.0 +b +12 +10 +8 +６ +4 +Sep-30 Oct-03 +Oct-06 0ct-09 Mar-06 +Mar-10 +Mar-14 Mar-28 Apr-01 +Apr-05 Apr-09 +Date (2019) +Date (2020) +Date (2020)9 +Figure 6. Panels a-c (red error bars) show the evolution of abundance in the logarithmic scale with A(H)=12 for Mg, Al, and Si +during the evolution of AR12749, AR12758 and AR12759. The magenta bars represented the average abundances when the ARs +were very bright, as bounded by the vertical dashed lines. The y-error bars represent 1σ uncertainty for each parameter, and the +x-error bars represent the duration over which a given spectrum is integrated. The black horizontal dashed lines represent the +average abundances for the quiet Sun in the absence of any AR reported by Vadawale et al. (2021b). XSM light curves for each +AR are shown in gray in the background, and the blue color on the XSM light curves represents the time duration excluding the +flaring activities. The range of coronal and photospheric abundances from various authors compiled in the CHIANTI database +are shown as orange and green bands. The right y-axis shows the FIP bias values for the respective elements with respect to +average photospheric abundances. +MK, and the EM increases as the emitting plasma vol- +ume increase until it has emerged completely. We also +derived the evolution of the FIP bias during this period, +shown in Figure 7d for Si. During this period, as the +emission from the Mg and Al line complex was weak +compared with the background solar emission, the de- +rived FIP bias for Mg and Al has a large uncertainty +and is not shown here. Within ∼10 hours of the AR +emergence, the FIP bias was already close to 3, and re- +mained almost constant throughout the evolution. So +the emerging hot core loops do not show any variation, +in agreement with previous suggestions. Recall that the +variations in FIP bias reported earlier (e.g., Widing and +Feldman 2001) were observed in the cool loops, not the +core loops. +5.4. Enhanced bias for Al +Figure 8 shows the average values of the FIP bias +(relative to the photospheric abundance Asplund et al. +(2009)) for all the elements as a function of their FIP +values. +The lower FIP element, Al (FIP = 5.99), is +found to have the highest FIP bias of 6-7, whereas the +low-FIP elements, Mg (FIP = 7.65) and Si (FIP = 8.15), +are found to have a lower FIP bias of ∼3. The mid/high +FIP element, S, is found to have a much lower FIP bias +of a factor of ∼ 1.5. A higher FIP bias for Al is note- +worthy and may point to an intriguing physical process. +However, this may also be a modeling artifact. +One of the possibilities could be due to missing flux +caused by the presence of multi-thermal plasma provid- +ing strong signals from emission lines of Al or Mg formed + +AR12749 +AR12758 +AR12759 +8.2 +a +3 +8.0 +2 +7.8 +7.6 +b +7.2 +6 +4 +bias +6.9 +3 +N +FIP +2 +6.6 +1 +6.3 +c +4 +8.0 +3 ++ +S +7.8 +7.6 +7.4 +Sep-30 ( +Oct-03 +Oct-06 Oct-09 Mar-06 Mar-10 +Mar-14 Mar-18Mar-28 Apr-01 Apr-05 Apr-09 +Date (2019) +Date (2020) +Date (2020)10 +Figure 7. Results showing the emerging phase of AR12758. +The black curve in panel a shows the evolution of the total +unsigned photospheric magnetic flux. Panel b and c show the +evolution of temperature and EM. Panel d shows the evolu- +tion of FIP bias for Si. The dashed lines in panels b-d repre- +sent the corresponding parameter for the background solar +emission from the rest of the solar-disk except AR. The back- +ground grey curves in each panel represent the X-ray light +curve observed by XSM. Whereas the blue curves represent +the selected times excluding the flaring period, representing +the quiescent AR. +at different temperatures. To verify this we have simu- +lated the emission lines in the energy range of the Mg/Al +line complex by considering the isothermal model and a +multi-thermal model using the AR DEM of AR12759, +reported by Del Zanna et al. (2022) (see Figure B.1 in +Appendix B). Similar line intensities from various ion- +ization stages of Al and Mg can be seen in both the +isothermal and multi-thermal models, confirming that +the absence of the flux is not the result of multi-thermal +plasma. +Another possibility is that missing flux is caused by +missing lines of Al or Mg (mostly satellite lines) that +are not yet present in CHIANTI version 10. We have +analysed the high-resolution spectroscopic observations +described by Walker et al. (1974) and found several ob- +served lines that are missing in the database. However, +the total missing flux, compared to the predicted flux +by CHIANTI is not enough to explain the anomalous +Al abundance. However, the Walker et al. (1974) obser- +vations were taken during a high level of solar activity, +so it is possible that the missing lines have a stronger +contribution at 3 MK. The Al abundance is currently +clearly overestimated by some degree. +Although this analysis is not conclusive enough to rule +out Al’s high FIP bias as an artifact, it is also not suf- +ficient to conclude that it is not real. A higher Al FIP +bias could be real. This might be explained by examin- +ing a few particular scenarios from the Ponderomotive +force model (Laming 2015) proposed by Laming (pri- +vate communication), which could be investigated in a +subsequent study. +We have also compared the AR core FIP bias obtained +with that of the different solar activity levels measured +by the XSM in previous research. These are overplot- +ted in Figure 8. +The blue points show the FIP bias +during the quiet Sun period, which is dominated by X- +ray Bright Points (XBP), as reported by Vadawale et al. +(2021b). While the green points depict the FIP bias dur- +ing the peak of the solar flares as reported by Mondal +et al. (2021). The FIP bias of the AR core (red points) +shows a consistently higher value for the elements Al, +Mg, and Si compared with the FIP bias of XBPs (green +points). Since ARs have substantially higher magnetic +activity than the XBPs, the increased FIP bias of the +ARs relative to the XBPs is expected from the Pondero- +motive force model. On the other hand, chromospheric +evaporation during the flaring mechanism results in a +near unit FIP bias during the peak of the flares (Mon- +dal et al. 2021). +6. SUMMARY +We present the evolution of plasma characteristics for +three ARs using disk-integrated soft X-ray spectroscopic +observations from the XSM to make simultaneous line +and continuum measurements. Carrying out a compre- +hensive study of an AR using the Sun-as-a-star mode +observation is challenging because of the presence of +multiple activities throughout the solar cycle. Unique + +a +Mx) +20 +(×1021 +10 +B-flux +b +4 +(MK) +3 +C +10 +5 +EM +0 +d +6 +4 +bias +FIP +2 +-20 +0 +20 +40 +Hours from 07-Mar-202011 +Table 1. Best fitted parameters for the average spectrum of each AR. +AR +T +EM +Mg +Al +Si +S +(MK) +(1046 cm−3) +12749 +3.14+0.04 +−0.05 +2.46+0.24 +−0.19 +8.00+0.02 +−0.03 +7.28+0.05 +−0.06 +8.00+0.02 +−0.02 +7.23+0.06 +−0.05 +12759 +3.22+0.04 +−0.02 +4.30+0.21 +−0.28 +7.95+0.02 +−0.02 +7.26+0.04 +−0.04 +8.04+0.01 +−0.01 +7.23+0.02 +−0.03 +12758 +2.99+0.05 +−0.03 +3.48+0.25 +−0.31 +7.95+0.03 +−0.02 +7.23+0.06 +−0.05 +8.02+0.02 +−0.02 +7.32+0.05 +−0.06 +6 +7 +8 +9 +10 +11 +FIP (eV) +2 +0 +2 +4 +6 +8 +FIP bias +Al +Mg +Si +S +AR +XBP +Flare +Figure 8. Variation of the FIP bias with the FIP of the ele- +ments. The red points are the averaged FIP bias for the ARs +reported in the present study. The blue points are the FIP +bias for the XBPs as reported by Vadawale et al. (2021b). +The green points are the measured FIP bias during the peak +of solar flares as reported by Mondal et al. (2021). +XSM observations made during the minimum of Solar +Cycle 24 allowed the study of the evolution of temper- +ature, EM, and the abundances of Mg, Al, and Si for +the individual ARs in the absence of any other notewor- +thy activity on the solar disk. Since the ARs were the +principal contributors of disk-integrated X-rays during +their evolution, the temperature and EM followed their +X-ray light curve. The average temperature of all the +AR is ∼3 MK, close to the well-known temperature of +the AR core. Irrespective of the activity and age of the +ARs, the abundances or the FIP biases of Al, Mg, and Si +were found to be consistently greater than their photo- +spheric values without much variation. The abundance +values develop within ∼10 hours of the appearance of +the AR during its emerging phase. Throughout the AR +evolution, the low FIP elements, Mg and Si, have a FIP +bias close to 3, whereas the mid-FIP element, S, has an +average FIP bias of ∼1.5. The lowest FIP element, Al, +has a greater FIP bias of ∼6-7. After discussing vari- +ous modeling artifacts, the Al abundance appears to be +overestimated, although the exact factor is unknown. +Increased Al abundance could be real, implying that +low-FIP elements degree of FIP bias is linked to their +FIP values. Future spectroscopic studies to measure the +FIP bias for more low-FIP elements (for example, Ca, +whose FIP bias is between Al and Mg) would help us +to better understand this phenomenon. In this regard, +recent and upcoming X-ray spectrometers (for example, +DAXSS: (Schwab et al. 2020) onboard INSPIRESat-1, +SoLEXS (Sankarasubramanian et al. 2011) onboard up- +coming Aditya-L1 observatory, and rocket-borne spec- +trometer MaGIXS (Champey et al. 2022)) will be use- +ful. +ACKNOWLEDGMENTS +We acknowledge the use of data from the Solar X- +ray Monitor (XSM) on board the Chandrayaan-2 mis- +sion of the Indian Space Research Organisation (ISRO), +archived at the Indian Space Science Data Centre +(ISSDC). The XSM was developed by the engineer- +ing team of Physical Research Laboratory (PRL) lead +by Dr. +M. Shanmugam, with support from various +ISRO centers. +We thank various facilities and the +technical teams from all contributing institutes and +Chandrayaan-2 project, mission operations, and ground +segment teams for their support. +Research at PRL +is supported by the Department of Space, Govt. +of +India. +We acknowledge the support from Royal So- +ciety through the international exchanges grant No. +IES\R2\170199. GDZ and HEM acknowledge support +from STFC (UK) via the consolidated grant to the +atomic astrophysics group at DAMTP, University of +Cambridge (ST\T000481\1). AB was the J C Bose Na- +tional Fellow during the period of this work. We thank +Dr. Martin Laming for the useful discussion on anoma- +lous Al abundance. +APPENDIX + +12 +A. RESULTS OF MCMC ANALYSIS +We carried out Markov Chain Monte Carlo (MCMC) analysis of the spectra to obtain the regions of parameter space +that best fits the observed spectra. This was done using the ‘chain’ method available within XSPEC. Figure A1 shows +the corner plot of the results for the spectrum on 01-Oct-2019. The results show that all parameters are well constrained +by the spectra. Particularly, we note that there is no anti-correlation observed between Al and Mg abundances showing +that the enhances Al abundances obtained cannot be adjusted by enhancements in Mg abundances. Similar trends +are observed for spectra of other days as well. +7.80 +7.85 +7.90 +7.95 +Mg +7.0 +7.1 +7.2 +7.3 +Al +7.92 +7.95 +7.98 +8.01 +Si +3.30 +3.36 +3.42 +3.48 +3.54 +T +3.5 +4.0 +4.5 +5.0 +5.5 +EM +7.80 +7.85 +7.90 +7.95 +Mg +7.0 +7.1 +7.2 +7.3 +Al +7.92 +7.95 +7.98 +8.01 +Si +3.5 +4.0 +4.5 +5.0 +5.5 +EM +Figure A.1. Corner plot depicting the results of MCMC analysis for the fitted spectrum on 01-Oct-2019. The histograms +depict the marginalized distribution associated with each parameter. The scatter-plots are overlaid with contours representing +1σ, 2σ, and 3σ levels to show correlations between all parameters. The best-fit parameters are represented by green lines. +B. SIMULATED SPECTRUM +To check the effect of temperatures on the Mg/Al line fluxes in the XSM energy range of 1.55 to 1.70 keV, we +have compared the simulated spectra in the same energy range by considering the isothermal and multi-thermal DEM +models. Figure B.1 shows the simulated 3 MK spectrum (blue) overplotted with the multithermal spectrum (red). +The isothermal spectrum is generated for an emission measure of 1027 cm−5. The multithermal spectrum is derived by + +13 +1.56 +1.58 +1.60 +1.62 +1.64 +1.66 +1.68 +1.70 +Energy (keV) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized intensity +Mg XI, Al XI-XII +Al XII +Mg XI +Mg XI +DEM +Isothermal +Figure B.1. Simulated spectra from CHIANTI v 10 in the energy range of Mg/Al line complex of XSM observed spectrum. +Solid blue curve show the multi-thermal spectrum and dashed orange curve shows the isothermal spectrum. +using the reported quiescent AR DEM by Del Zanna et al. (2022), which was obtained from the Hinode EIS observation +of AR12759. For the comparison of both spectra, we have normalized them with the corresponding line flux of Mg XI, +and Al XI-XII. Similar line intensities predicted by both isothermal and multithermal models indicates that spectra +are insensitive to temperature in this case. + +14 +REFERENCES +Arnaud, K., Dorman, B., and Gordon, C. (1999). XSPEC: +An X-ray spectral fitting package. +Asplund, M., Grevesse, N., Sauval, A. J., and Scott, P. +(2009). The Chemical Composition of the Sun. +ARA&A, 47(1):481–522. +Baker, D., Brooks, D. H., D´emoulin, P., van +Driel-Gesztelyi, L., Green, L. M., Steed, K., and Carlyle, +J. (2013). Plasma Composition in a Sigmoidal Anemone +Active Region. ApJ, 778(1):69. +Baker, D., Brooks, D. H., D´emoulin, P., Yardley, S. L., van +Driel-Gesztelyi, L., Long, D. M., and Green, L. M. +(2015). FIP Bias Evolution in a Decaying Active Region. +ApJ, 802(2):104. +Baker, D., Brooks, D. H., van Driel-Gesztelyi, L., James, +A. W., D´emoulin, P., Long, D. M., Warren, H. P., and +Williams, D. R. (2018). Coronal Elemental Abundances +in Solar Emerging Flux Regions. ApJ, 856(1):71. +Bochsler, P. (2007). Minor ions in the solar wind. +A&A Rv, 14(1):1–40. +Brooks, D. H., Baker, D., van Driel-Gesztelyi, L., and +Warren, H. P. (2017). A solar cycle correlation of +coronal element abundances in sun-as-a-star +observations. Nature Communications, 8(1). +Brooks, D. H., Ugarte-Urra, I., and Warren, H. P. (2015). +Full-Sun observations for identifying the source of the +slow solar wind. Nature Communications, 6:5947. +Brooks, D. H. and Warren, H. P. (2011). Establishing a +Connection Between Active Region Outflows and the +Solar Wind: Abundance Measurements with EIS/Hinode. +ApJL, 727(1):L13. +Champey, P. R., Winebarger, A. R., Kobayashi, K., +Athiray, P. S., Hertz, E., Savage, S., Beabout, B., +Beabout, D., Broadway, D., Bruccoleri, A. R., Cheimets, +P., Davis, J., Duffy, J., Golub, L., Gregory, D. A., +Griffith, C., Haight, H., Heilmann, R. K., Hogue, B., +Hohl, J., Hyde, D., Kegley, J., Kolodzieczjak, J., +Ramsey, B., Ranganathan, J., Robertson, B., +Schattenburg, M. L., Speegle, C. O., Vigil, G., Walsh, R., +Weddenorf, B., and Wright, E. (2022). The Marshall +Grazing Incidence X-ray Spectrometer (MaGIXS). +Journal of Astronomical Instrumentation, 11(2):2250010. +Dahlburg, R. B., Laming, J. M., Taylor, B. D., and +Obenschain, K. (2016). PONDEROMOTIVE +ACCELERATION IN CORONAL LOOPS. The +Astrophysical Journal, 831(2):160. +Del Zanna, G. (2003). Solar active regions: The footpoints +of 1 MK loops. A&A, 406:L5–L8. +Del Zanna, G. (2012). Benchmarking atomic data for the +CHIANTI atomic database: coronal lines observed by +Hinode EIS. A&A, 537:A38. +Del Zanna, G. (2013). The multi-thermal emission in solar +active regions. A&A, 558:A73. +Del Zanna, G. (2019). The EUV spectrum of the Sun: +Quiet- and active-Sun irradiances and chemical +composition. A&A, 624:A36. +Del Zanna, G. and Mason, H. E. (2003). Solar active +regions: SOHO/CDS and TRACE observations of +quiescent coronal loops. A&A, 406:1089–1103. +Del Zanna, G. and Mason, H. E. (2014). Elemental +abundances and temperatures of quiescent solar active +region cores from X-ray observations. A&A, 565:A14. +Del Zanna, G. and Mason, H. E. (2018). Solar UV and +X-ray spectral diagnostics. Living Reviews in Solar +Physics, 15(1):5. +Del Zanna, G., Mondal, B., Rao, Y. K., Mithun, N. P. S., +Vadawale, S. V., Reeves, K. K., Mason, H. E., Sarkar, A., +Janardhan, P., and Bhardwaj, A. (2022). +Multiwavelength Observations by XSM, Hinode, and +SDO of an Active Region. Chemical Abundances and +Temperatures. ApJ, 934(2):159. +Doschek, G. A. and Warren, H. P. (2019). The Variability +of Solar Coronal Abundances in Active Regions and the +Quiet Sun. ApJ, 884(2):158. +Dwivedi, B. N., Curdt, W., and Wilhelm, K. (1999). +Analysis of Extreme-Ultraviolet Off-Limb Spectra +Obtained with SUMER/SOHO: Ne VI-Mg VI Emission +Lines. ApJ, 517(1):516–525. +Feldman, U. (1992). Elemental abundances in the upper +solar atmosphere. PhyS, 46(3):202–220. +Feldman, U., Landi, E., Doschek, G. A., Dammasch, I., and +Curdt, W. (2003). Free-Free Emission in the +Far-Ultraviolet Spectral Range: A Resource for +Diagnosing Solar and Stellar Flare Plasmas. ApJ, +593(2):1226–1241. +Feldman, U., Schuhle, U., Widing, K. G., and Laming, +J. M. (1998). Coronal composition above the solar +equator and the north pole as determined from spectra +acquired by the SUMER instrument onSOHO. The +Astrophysical Journal, 505(2):999–1006. +Feldman, U. and Widing, K. G. (2002). A review of the +first ionization potential effect on elemental abundances +in the solar corona and in flares. Physics of Plasmas, +9(2):629–635. +Fludra, A. and Schmelz, J. T. (1999). The absolute coronal +abundances of sulfur, calcium, and iron from +Yohkoh-BCS flare spectra. A&A, 348:286–294. + +15 +Freeland, S. L. and Handy, B. N. (1998). Data Analysis +with the SolarSoft System. SoPh, 182(2):497–500. +Gloeckler, G. and Geiss, J. (1989). The abundances of +elements and isotopes in the solar wind. In Waddington, +C. J., editor, Cosmic Abundances of Matter, volume 183 +of American Institute of Physics Conference Series, pages +49–71. +Ko, Y.-K., Young, P. R., Muglach, K., Warren, H. P., and +Ugarte-Urra, I. (2016). Correlation of Coronal Plasma +Properties and Solar Magnetic Field in a Decaying +Active Region. ApJ, 826(2):126. +Lakshitha, N., Mondal, B., Narendranath, S., and Paul, K. +(2022). Elemental abundances during A-class solar +flares: Soft X-ray spectroscopy from Chandrayaan-2 +XSM. Under preparation. +Laming, J. M. (2004). A Unified Picture of the First +Ionization Potential and Inverse First Ionization +Potential Effects. ApJ, 614(2):1063–1072. +Laming, J. M. (2009). Non-Wkb Models of the First +Ionization Potential Effect: Implications for Solar +Coronal Heating and the Coronal Helium and Neon +Abundances. ApJ, 695(2):954–969. +Laming, J. M. (2012). Non-WKB Models of the First +Ionization Potential Effect: The Role of Slow Mode +Waves. ApJ, 744(2):115. +Laming, J. M. (2015). The fip and inverse fip effects in +solar and stellar coronae. Living Reviews in Solar +Physics, 12:1–76. +Laming, J. M. (2017). The first ionization potential effect +from the ponderomotive force: On the polarization and +coronal origin of alfv´en waves. The Astrophysical +Journal, 844(2):153. +Mihailescu, T., Baker, D., Green, L. M., van +Driel-Gesztelyi, L., Long, D. M., Brooks, D. H., and To, +A. S. H. (2022). What Determines Active Region +Coronal Plasma Composition? ApJ, 933(2):245. +Mithun, N., Vadawale, S., Patel, A., Shanmugam, M., +Chakrabarty, D., Konar, P., Sarvaiya, T., Padia, G., +Sarkar, A., Kumar, P., Jangid, P., Sarda, A., Shah, M., +and Bhardwaj, A. (2021). Data processing software for +chandrayaan-2 solar x-ray monitor. Astronomy and +Computing, 34:100449. +Mithun, N. P. S., Vadawale, S. V., Sarkar, A., Shanmugam, +M., Patel, A. R., Mondal, B., Joshi, B., Janardhan, P., +Adalja, H. L., Goyal, S. K., Ladiya, T., Tiwari, N. K., +Singh, N., Kumar, S., Tiwari, M. K., Modi, M. H., and +Bhardwaj, A. (2020). Solar X-Ray Monitor on Board the +Chandrayaan-2 Orbiter: In-Flight Performance and +Science Prospects. SoPh, 295(10):139. +Mithun, N. P. S., Vadawale, S. V., Zanna, G. D., Rao, +Y. K., Joshi, B., Sarkar, A., Mondal, B., Janardhan, P., +Bhardwaj, A., and Mason, H. E. (2022). Soft X-Ray +Spectral Diagnostics of Multithermal Plasma in Solar +Flares with Chandrayaan-2 XSM. ApJ, 939(2):112. +Mondal, B., Klimchuk, J. A., Vadawale, S. V., Sarkar, A., +Zanna, G. D., Athiray, P. S., Mithun, N., Mason, H. E., +and Bhardwaj, A. (2022). Role of small-scale impulsive +events in heating the X-ray bright points of the quiet +Sun. Submitted to ApJ. +Mondal, B., Sarkar, A., Vadawale, S. V., Mithun, N. P. S., +Janardhan, P., Del Zanna, G., Mason, H. E., +Mitra-Kraev, U., and Narendranath, S. (2021). +Evolution of Elemental Abundances during B-Class Solar +Flares: Soft X-Ray Spectral Measurements with +Chandrayaan-2 XSM. ApJ, 920(1):4. +Pottasch, S. R. (1963). The Lower Solar Corona: +Interpretation of the Ultraviolet Spectrum. ApJ, 137:945. +Saba, J. L. R. and Strong, K. T. (1993). Coronal +abundances of O, Ne, Mg, and Fe in solar active regions. +Advances in Space Research, 13(9):391–394. +Sankarasubramanian, K., Ramadevi, M. C., Bug, M., +Umapathy, C. N., Seetha, S., Sreekumar, P., and Kumar +(2011). SoLEXS - A low energy X-ray spectrometer for +solar coronal studies. In Astronomical Society of India +Conference Series, volume 2 of Astronomical Society of +India Conference Series, pages 63–69. +Schwab, B. D., Sewell, R. H. A., Woods, T. N., Caspi, A., +Mason, J. P., and Moore, C. (2020). Soft X-Ray +Observations of Quiescent Solar Active Regions Using +the Novel Dual-zone Aperture X-Ray Solar Spectrometer. +ApJ, 904(1):20. +Shanmugam, M., Vadawale, S. V., Patel, A. R., Adalaja, +H. K., Mithun, N. P. S., Ladiya, T., Goyal, S. K., Tiwari, +N. K., Singh, N., Kumar, S., Painkra, D. K., Acharya, +Y. B., Bhardwaj, A., Hait, A. K., Patinge, A., Kapoor, +A. h., Kumar, H. N. S., Satya, N., Saxena, G., and +Arvind, K. (2020). Solar X-ray Monitor Onboard +Chandrayaan-2 Orbiter. Current Science, 118(1):45–52. +Sheeley, N. R., J. (1995). A Volcanic Origin for High-FIP +Material in the Solar Atmosphere. ApJ, 440:884. +Sheeley, N. R., J. (1996). Elemental Abundance Variations +in the Solar Atmosphere. ApJ, 469:423. +Testa, P., Martinez-Sykora, J., and De Pontieu, B. (2022). +Coronal Abundances in an Active Region: Evolution and +Underlying Chromospheric and Transition Region +Properties. arXiv e-prints, page arXiv:2211.07755. + +16 +Vadawale, S., Shanmugam, M., Acharya, Y., Patel, A., +Goyal, S., Shah, B., Hait, A., Patinge, A., and +Subrahmanyam, D. (2014). Solar x-ray monitor (xsm) +on-board chandrayaan-2 orbiter. Advances in Space +Research, 54(10):2021 – 2028. Lunar Science and +Exploration. +Vadawale, S. V., Mithun, N. P. S., Mondal, B., Sarkar, A., +Janardhan, P., Joshi, B., Bhardwaj, A., Shanmugam, M., +Patel, A. R., Adalja, H. K. L., Goyal, S. K., Ladiya, T., +Tiwari, N. K., Singh, N., and Kumar, S. (2021a). +Observations of the quiet sun during the deepest solar +minimum of the past century with chandrayaan-2 XSM: +Sub-a-class microflares outside active regions. The +Astrophysical Journal Letters, 912(1):L13. +Vadawale, S. V., Mondal, B., Mithun, N. P. S., Sarkar, A., +Janardhan, P., Joshi, B., Bhardwaj, A., Shanmugam, M., +Patel, A. R., Adalja, H. K. L., Goyal, S. K., Ladiya, T., +Tiwari, N. K., Singh, N., and Kumar, S. (2021b). +Observations of the quiet sun during the deepest solar +minimum of the past century with chandrayaan-2 XSM: +Elemental abundances in the quiescent corona. The +Astrophysical Journal Letters, 912(1):L12. +Walker, A. B. C., J., Rugge, H. R., and Weiss, K. (1974). +Relative Coronal Abundances Derived from X-Ray +Observations. I. Sodium, Magnesium, Aluminum, Silicon, +Sulfur, and Argon. ApJ, 188:423–440. +Widing, K. G. (1997). Emerging Active Regions on the +Sun and the Photospheric Abundance of Neon. ApJ, +480(1):400–405. +Widing, K. G. and Feldman, U. (1993). Nonphotospheric +abundances in a solar active region. ApJ, 416:392. +Widing, K. G. and Feldman, U. (2001). On the Rate of +Abundance Modifications versus Time in Active Region +Plasmas. ApJ, 555(1):426–434. +Winebarger, A. R., Warren, H. P., Schmelz, J. T., Cirtain, +J., Mulu-Moore, F., Golub, L., and Kobayashi, K. (2012). +Defining the “Blind Spot” of Hinode EIS and XRT +Temperature Measurements. ApJL, 746(2):L17. +Young, P. R. and Mason, H. E. (1997). The Mg/Ne +abundance ratio in a recently emerged flux region +observed by CDS. SoPh, 175(2):523–539. + diff --git a/4NE1T4oBgHgl3EQf6AXQ/content/tmp_files/load_file.txt b/4NE1T4oBgHgl3EQf6AXQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..03ee8a935a2d3ed922f8cd338bfecb9ae9123580 --- /dev/null +++ b/4NE1T4oBgHgl3EQf6AXQ/content/tmp_files/load_file.txt @@ -0,0 +1,1217 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf,len=1216 +page_content='Draft version January 10, 2023 Typeset using LATEX twocolumn style in AASTeX63 Evolution of elemental abundances in hot active region cores from Chandrayaan-2 XSM observations Biswajit Mondal,1, 2 Santosh V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Vadawale,1 Giulio Del Zanna,3 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Mithun,1 Aveek Sarkar,1 Helen E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Mason,3 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Janardhan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='1 and Anil Bhardwaj1 1Physical Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Navrangpura,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Ahmedabad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Gujarat-380 009,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' India 2Indian Institute of Technology Gandhinagar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Palaj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Gandhinagar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Gujarat-382 355,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' India 3DAMTP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Centre for Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Wilberforce Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Cambridge CB3 0WA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' UK ABSTRACT The First Ionization Potential (FIP) bias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' whereby elemental abundances for low FIP elements in different coronal structures vary from their photospheric values and may also vary with time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' has been widely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In order to study the temporal variation, and to understand the physical mechanisms giving rise to the FIP bias, we have investigated the hot cores of three ARs using disk-integrated soft X-ray spectroscopic observation with the Solar X-ray Monitor (XSM) onboard Chandrayaan-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Observations for periods when only one AR was present on the solar disk were used so as to ensure that the AR was the principal contributor to the total X-ray intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The average values of temperature and EM were ∼3 MK and 3×1046 cm−3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Regardless of the age and activity of the AR, the elemental abundances of the low FIP elements, Al, Mg, and Si were consistently higher than their photospheric values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The average FIP bias for Mg and Si was ∼3, whereas the FIP bias for the mid-FIP element, S, was ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' However, the FIP bias for the lowest FIP element, Al, was observed to be higher than 3, which, if real, suggests a dependence of the FIP bias of low FIP elements on their FIP value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Another major result from our analysis is that the FIP bias of these elements is established in within ∼10 hours of emergence of the AR and then remains almost constant throughout its lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Keywords: Solar X-ray corona, Solar abundances, FIP bias, FIP effect, Active Region 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' INTRODUCTION The earlier study of the Sun as a star by Pottasch (1963) revealed that solar coronal abundances are dif- ferent from those of the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The differences are correlated to the First Ionization Potential (FIP) of the element, in the sense that the abundance ratio of a low-FIP (less than 10 eV) element versus that of a high-FIP element is higher in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A measure of the difference is the so called FIP bias, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' the ratio between the coronal and the photospheric abundance of an element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In most of the available literature, the FIP bias has been (and still is) estimated by measuring the relative abundances between elements, and not relative to hy- drogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' This is due to the fact that abundance mea- surements with respect to Hydrogen in the low corona, and on-disk is non-trivial, due to the lack of H-emission Corresponding author: Biswajit Mondal biswajit70mondal94@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='com, biswajitm@prl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='in lines at a few million Kelvin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Hence, whether it is the low-FIP elements that have an increased abundance or the high-FIP elements that have a reduced one (com- pared to their photospheric values) has been a subject of continued debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Further, it has become clear that different solar struc- tures have different FIP biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' There are also indica- tions that the FIP bias depends on the temperature of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' For a long time, it has been widely accepted that coronal abundances in active regions increase with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We refer the reader to the recent reviews by Lam- ing (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna and Mason (2018) for more de- tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We also provide in the following section a brief summary of available measurements related to active re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Knowledge of the elemental abundances in different atmospheric layers of the Sun is a topic of great inter- est to the solar physics community mainly due to the following two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The first is that they provide, in principle, a way to link the solar source regions to the various components of the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In fact, elemental abundance variations are also clearly observed in-situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='03519v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='SR] 9 Jan 2023 2 The slow-speed solar wind has a high FIP bias simi- lar to that measured in AR core loops, 3MK, whereas the high-speed wind has a near unit FIP bias, similar to that of coronal holes (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Brooks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Gloeckler and Geiss 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Bochsler 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Brooks and Warren 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The second reason is that studying abundance vari- ations might contribute to a better understanding of the physical processes at play in the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In fact, we know that the FIP bias is closely related to the magnetic field activity of the Sun (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Feld- man and Widing 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Brooks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Ponderomotive force model (Laming 2004, 2009, 2012, 2017) is now widely accepted, as it is able to reproduce the main characteristics of the FIP effect, as measured in-situ and remotely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' According to this model, the separation of ions from neutral atoms within closed loops in an upward direction is caused by the reflec- tion of downward propagating Alfv’en waves at chromo- spheric heights, causing an enhancement of the low-FIP elements in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Since coronal waves can be pro- duced by mechanisms that heat the solar corona, it is thought that the mechanism underlying the FIP effect is inextricably linked to processes that heat the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Hence, measuring the FIP bias is an impor- tant diagnostic for coronal plasma characteristics (Lam- ing 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Dahlburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In this paper, we focus on the elemental abundances of hot, quiescent AR core emission at 3 MK, by provid- ing line-to-continuum measurements of the Sun in the soft X-ray energy band using data from the Solar X-ray Monitor (XSM: Vadawale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Shanmugam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' It may be noted here that the XSM is the only spectrometer to have observed the Sun in the 1-15 keV range during the minimum of solar cycle 24 with an en- ergy resolution better than 180 eV at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='9 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' This reso- lution is sufficient to measure the abundances of several elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The soft X-ray continuum is dominated by free-free radiation (with some free-bound emission, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Figure 12b of Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2021), which primarily originates from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Hence, measuring the abundances of an emission line with respect to the continuum provides the absolute abundance of that element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' It should be noted that the measurement of free-free emission can also be carried out in the EUV energy band, but it is limited to large flares (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The XSM energy band is sensitive to temperatures above 2 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' When the Sun was at minimum activity levels, without any ARs, the XSM observed a steady sig- nal originating from X-ray Bright Points (XBPs), with a peak emission around 2 MK (Vadawale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' When a single non-flaring AR is present, the signal is dominated by the AR’s near-isothermal ∼ 3 MK emis- sion (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' This provides an ex- cellent opportunity to measure the FIP bias of the hot quiescent core for individual active regions during their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In the literature, very few abundance measurements are know to be associated specifically with the 3 MK emission from quiescent AR cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' These are sum- marised in Del Zanna and Mason (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' X-ray spectra in the 10–20 ˚A range have provided the relative abun- dances of the low-FIP Fe, Mg vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' O, Ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Most stud- ies provided results on single active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Saba and Strong (1993) reported a significant variability of the FIP bias using SMM/FCS observations of several active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' On the other hand, a re-analysis of several qui- escent AR cores with improved atomic data and using a multi-thermal DEM technique by Del Zanna and Ma- son (2014) indicated the same FIP bias, around 3, for all active regions, irrespective of their age and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Since 2006, EUV spectra from the Hinode EIS instru- ment have provided an opportunity to measure the rel- ative FIP bias between low-FIP elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Fe, Si) and the high-FIP Ar, as well as the mid-FIP S, which actually shows the same abundance variations as the high-FIP elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' An example case was discussed by Del Zanna (2013), showing that the FIP bias in the EUV of 3 MK plasma was the same as in the X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Con- sidering the size of the emitting plasma and its emission measure, Del Zanna (2013) concluded that it should be the low-FIP elements that are over-abundant by about a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022) carried out a multi-wavelength study of an AR as it crossed the solar disk which was ob- served by XSM as well as by SDO/AIA, Hinode/EIS and Hinode/XRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The relative FIP bias obtained from Hin- ode/EIS observations confirmed the Del Zanna (2013) results, and showed no variation with the disk passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The analysis of simultaneous XSM spectra on two days also indicated no significant variability, and provided an absolute FIP bias for Si of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' close to the value suggested by Del Zanna (2013), and also very close to the prediction of Laming’s model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In the present study, we extend the previous XSM analysis to all the quiescent periods of the same active region, and also investigate two other active regions dur- ing their disk crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' One AR in particular is of in- terest as it emerged on-disk, and hence offers the op- portunity to study the elemental abundances during the early phase of the evolution of an AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The rest of the paper is organized as follows: Sec- tion 2 provides a short overview of previous abundance measurements in active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Section 3 describes the 3 observations and data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Section 4 provides a detailed spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' After obtaining the results, these are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Section 6 provides a brief summary of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' HISTORICAL OVERVIEW Spatially resolved measurements of the relative FIP bias have been carried out by several authors (see,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Widing and Feldman 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Sheeley 1995, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Widing 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Widing and Feldman 2001) using Skylab spectro- heliograms with Mg, Ne transition region lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' These are formed well below 1 MK, in the legs of active re- gion ‘cool’ (1 MK) loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' They found photospheric com- position (FIP bias=1) for newly emerged closed loops, but increasing FIP bias reaching a value of 3-4 within a timescale of 1-2 days (Widing and Feldman 2001), and much higher values, up to about 10, within a few more days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Differing FIP biases were also obtained by Young and Mason (1997) and Dwivedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1999) using Mg and Ne line ratios observed by the CDS and SUMER spectrometers onboard the Solar and Heliospheric Ob- servatory (SOHO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The large values are hard to reconcile with in-situ measurements, where the FIP bias is at most 3, and also with theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' However, Del Zanna (2003) pointed out that as the cool AR loops are almost isothermal in their cross-section, the assumption that a smooth emis- sion measure distribution was present in the plasma, used to interpret the Skylab data, was not justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna (2003) took the intensities measured by Wid- ing and Feldman (1993), and using an emission measure loci approach, showed that a FIP bias of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='7 was con- sistent with the data, much lower than the value of 14 reported by Widing and Feldman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna (2003) also analysed the legs of several cool loops observed by SoHO/CDS and found photospheric abundances, al- though a similar analysis for other loops by Del Zanna and Mason (2003) found a FIP bias of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In summary, the legs of cool AR loops do show a range of FIP bias values, between 1 and 4, and perhaps occa- sionally larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' However, the very high FIP biases found from Skylab data were largely overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' As shown by Del Zanna and Mason (2003), active re- gion cores are composed not only of cool 1 MK loops and unresolved, almost isothermal 3 MK loops, but also unresolved emission in the 1–3 MK range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The plasma at different temperatures is generally not cospatial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' There is evidence from Hinode EIS observations of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Si X, S X lines that this ≃2 MK emission has a lower relative FIP bias, around 2 (see,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Further studies using the same lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2013, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Doschek and Warren 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Mihailescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Ko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Testa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2022) have shown some variation (around the value of 2) of the relative FIP bias within each active region, but little variability in time, except during the decay phase, when an AR effectively disappears and the relative abundances become photo- spheric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In summary, active region structures formed at tem- peratures below 2 MK show a range of relative FIP bi- ases, and some temporal variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The few observa- tions of the hotter, 3 MK, AR cores have in contrast shown a remarkable consistency, with relative FIP bi- ases around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Finally, to interpret observations of the Sun as a star, one needs to take into account the above (and other) issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' As shown by Del Zanna (2019), when the Sun’s actvity is at a minimum with no active region present on the solar disk, the corona around 1 MK shows near photospheric abundances, whereas in presence of active regions, the FIP bias for the 1 MK emission stays the same, but the hotter emission shows a higher relative FIP bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' When active regions flare, the high tem- perature plasma shows nearly photospheric composition around the peak X-ray emission (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' OBSERVATIONS AND DATA ANALYSIS Observations of the Sun were carried out with the XSM during the minimum of solar cycle 24, when no active regions were present, covering the years 2019- 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Results are given in Vadawale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' They reported intermediate abundances of low-FIP elements (Mg, Al, and Si) of 2 MK plasma, primarily originating from X-ray Bright Points, XBPs (Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Frequent micro-flaring activity was observed and found to be occurring everywhere on the solar disk, even when no ARs were present (Vadawale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' During the minimum of solar cycle 24, XSM observed the disk passage of a few individual, isolated ARs in the absence of any other major activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' When ARs were present on-disk, XSM recorded hundreds of small flares of dif- ferent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Elemental abundance variations during these small flares were found, for the first time, to ini- tially drop to photospheric values, then rapidly return to coronal values, as described by Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021), Mithun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022), and Lakshitha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In this paper, we analyze the temporal evolution of active regions outside of flaring activity and for this we have chosen to study three isolated active regions: AR12749, AR12758, and AR12759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' XSM data contain spectra at 1 s cadence in a raw (level-1) daily file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Since the visibility of the Sun varies within the XSM field-of-view (FOV), with the Sun be- 4 ing sometimes outside the FOV or being occulted by the Moon, the data include both solar and non-solar spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The XSM Data Analysis Software (XSMDAS: Mithun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021)) has been used to generate the level-2 sci- ence data product using the appropriate Good Time Intervals (GTIs) and the other necessary instrumental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The available default level-2 data contains the effective area corrected light curves for every second and spectra for every minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' XSMDAS also provides the functionality to generate the light curves and spec- tra for a given cadence and energy range, which we have used in the present analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Using the XSMDAS, we have generated 2 min av- eraged XSM light curves in the energy range of 1-15 keV during the disk passage of the AR12749, AR12758, and AR12759, as shown in the three panels of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' During the evolution of these three ARs, representative full disk X-ray images taken by the XRT Be-thin fil- ter are shown in the top row of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' AR12749 (Figure 1a) appeared from the east limb on Sept 29, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Whilst crossing the solar disk, it became fainter towards the west limb and went behind the limb on 14 Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' AR12758 (Figure 1b) appears to form on disk on 06 Mar 2020 and fully emerged after 08 Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' It decays whilst crossing the solar disk and finally goes behind the west limb on 18 Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' AR12759 appeared from the east limb on 29 Mar 2020 and transited the solar disk until 14 Apr 2020, before disappearing behind the west limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The full disk XRT images show that during the pas- sage of these three ARs, no other major activity was present on the solar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Thus, we conclude that these three ARs were primarily responsible, during their disk passage, for the enhanced X-ray emission observed by the XSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' These ARs produced many small B/A-class flares, seen as multiple spikes in the XSM light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Detailed studies of these small flares were reported by Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021) and Lakshitha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In the present study, we have selected only the quies- cent periods from the observed light curves by exclud- ing the periods when the small flares occurred using a semi-automated graphical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' For example, Fig- ure 2 shows the representative selection (orange shaded regions) for the AR quiescent durations on 2020-04-06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' These identified time intervals were used as user-defined GTIs to generate the spectra for quiescent ARs on a daily basis in order to carry out the detailed spectral analysis as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' SPECTRAL ANALYSIS Broad-band soft X-ray spectra of the solar corona con- sist of a continuum as well as the emission lines of the different elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Modeling the soft X-ray spectrum provides the measurements of the temperature, emission measure, and elemental abundances (with respect to hy- drogen) of the emitting plasma (Del Zanna and Mason 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We use the chisoth model (Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2021) for the spectral fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The chisoth is a local model of the X-ray spectral fitting package (XSPEC: Arnaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1999)), and it estimates the theoretical spectrum using the CHIANTI atomic database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' It takes temper- ature, emission measure (EM: which is related to the density of the plasma), and the elemental abundances of the elements from Z=2 to Z=30 as free variables for the spectral fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' After generating the spectra for the quiescent peri- ods, we fitted them with an isothermal emission model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' For the spectral fitting, we ignored the spectra below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='3 keV where the XSM response is not well-known (Mithun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2020), and above the energy where the solar spec- trum is dominated by the non-solar background spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' During the spectral fitting, the temperature, EM, along with the abundances of Mg, Al, and Si (whose emission lines are prominent in the XSM spectrum) were kept as variable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The 1σ uncertainty of each free parameter was also estimated using the standard procedure in XSPEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Although the S line complex is visible in the spectra, including it in the spectral fits as a free parameter causes a large uncertainty in the measurement of the S abun- dance because of its poor statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Hence, we fixed the S abundances along with the abundances of other elements (whose emission lines are not visible in the ob- served spectra) with the coronal abundances of Feldman (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' However, we found that the measurement of the S abundance is possible for the summed spectrum of the entire AR period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Figure 3 shows the representative XSM spectra, for the three ARs fitted, in different colours, with an isother- mal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The points with error bars represent the observed spectra, whereas the solid curves represent the best-fit modeled spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The grey error bars represent the non-solar background spectrum, which is subtracted from the observed spectra during the spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The lower panel shows the residual between the observed and model spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We have fitted all the spectra in a similar way and found that all of them are well described by isothermal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The X-rays observed by XSM originated from both the AR and the background quiet Sun regions (outside the AR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' To determine how much emission is due to the background quiet Sun regions, we estimate the average quiet Sun spectrum using an average quiet-Sun temper- ature, EM, and abundances, as reported by Vadawale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The average quiet Sun spectrum is shown 5 Sep-29 Oct-01 Oct-03 Oct-05 Oct-07 Oct-09 Oct-11 Date (2019) 101 102 103 XSM Counts (s 1) AR12749 a Mar-06 Mar-08 Mar-10 Mar-12 Mar-14 Mar-16 Mar-18 Date (2020) 101 102 103 XSM Counts (s 1) AR12758 b Mar-26 Mar-28 Mar-30 Apr-01 Apr-03 Apr-05 Apr-07 Apr-09 Apr-11 Apr-13 Date (2020) 101 102 103 XSM Counts (s 1) AR12759 c Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' XSM 1-15 keV light curves during the disk passage of AR12749 (panel a), AR12758 (panel b) and AR12759 (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The top row of each panel shows representative full disk X-ray images (negative intensities) taken with the XRT Be-thin filter during the evolution of the ARs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The vertical dashed lines represent the timing of the XRT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 6 05:33 11:06 16:40 22:13 hh:mm on 2020-04-05 100 101 102 103 Rate(c/s) c 05:33 11:06 16:40 22:13 hh:mm on 2020-03-11 100 101 102 103 Rate(c/s) b 05:33 11:06 16:40 22:13 hh:mm on 2019-10-01 100 101 102 103 Rate(c/s) a Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Selection of the quiescent AR periods (orange- shaded regions) from the XSM light-curves for one represen- tative day of AR12749 (panel a), AR12758 (panel b), and AR12759 (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' by the black dashed curve in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The quiet Sun spectrum is found to be almost an order of magnitude lower than the spectra of the active period when the ARs were very bright on the solar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We thus con- clude that the X-ray emission of the active periods is primarily dominated by the AR emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Separating the AR emission from the background quiet Sun emission would be possible by subtracting the quiet-sun spectra from the AR spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' But, as the ef- fective area of the XSM varies with time, this is not recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' It is possible to model the AR spectra using a two-temperature (2T) component model rather than subtracting the quiet Sun spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' This is what we have chosen to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' One temperature corresponds to the background solar emission originating from the regions outside the AR and the second temperature corresponds to the AR plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We have modeled a few AR spec- tra with a two-temperature (2T) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' During the 2T spectral fitting, the parameters of the background solar emission were kept fixed to the average quiet-Sun values reported by Vadawale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' For the AR compo- nent, the temperature, EM, along with the abundances of Mg, Al, and Si, were kept as variable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We found that the 2T model can describe the XSM spectra for the active periods with similar best-fitted parameters as those obtained by the isothermal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' This verifies that the AR emission dominates the spectra of the AR periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Thus, in this study, we show the results of the isothermal analysis in Figure 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' This is discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' It is interesting to study how the plasma parameters vary during the emerging phase of the AR12758, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', from 07-Mar-2020 to 09-Mar-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Figure 4 shows the evolution of the photospheric magnetograms (top row) and the X-ray emission (bottom row) as observed by SDO/HMI and the Be-thin filter of Hinode/XRT re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' These images were created by de-rotating the synoptic data of HMI1 and XRT2 to a common date (08-Mar-2020) using the standard procedure of Solar- SoftWare (SSW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Freeland and Handy 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We also determined the total unsigned photospheric magnetic flux for the regions ±10 G within the field-of-view shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' During this emerging flux period, we car- ried out a time-resolved spectroscopic study using the XSM observations with finer time bins of less than a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' However, during this period, as the emission from the AR was not very bright, the emission from the AR and the rest of the Sun could be mixed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Thus to derive the evolution of the plasma parameters during this period, we modeled the observed XSM spectra with a 2T model, where one component represents the emis- sion from the AR, and the other represents the emission from the rest of the Sun, as discussed in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The results are shown in Figure 7 and dis- cussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' RESULTS AND DISCUSSION In this study, we have performed the X-ray spectral analysis for the evolution of three ARs as observed by the XSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The AR spectra (Figure 3) show a clear sig- nature of the thermal X-ray emission from the line com- 1 http://jsoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='edu/data/hmi/synoptic/ 2 http://solar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='montana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='edu/HINODE/XRT/SCIA/ 7 10 2 10 1 100 101 Counts (s 1keV 1) Mg Mg / Al Si Si S Quiet Sun AR12749 (Oct-01) AR12758 (Mar-11) AR12759 (Apr-05) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='25 Energy (keV) 5 0 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Soft X-ray spectra measured by the XSM for three representative days of the AR period are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Solid lines represent the best-fit isothermal model, and the residuals are shown in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Gray points correspond to the non-solar background spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' plexes of Mg, Al, Si, and S, along with the continuum emission up to ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The red points in Figure 5 show the evolution of the temperature and EM through- out the evolution of the three ARs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Figure 6 shows the evolution of abundances of Mg (panel a), Al (panel b), and Si (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The error bars associated with all the parameters along the y-axis represent the 1σ uncertain- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We also derived the average S abundance along with the other elements from the summed spectrum for the duration when the ARs were very bright on the solar disk (bounded by the vertical dashed lines in Figures 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' This provides the average parameters associ- ated with each AR, as shown by magenta bars and also given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The primary findings of the paper are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Temperature and emission measure Temperatures (T) and emission measures (EM) are close to the quiet Sun levels (black dashed lines in Fig- ure 5) when the ARs were absent from the solar disc or only partially present, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', 30 September 2019 and 6 March 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Once the ARs appear, the temperature rises to more than ∼3 MK from the ∼2 MK of the quiet Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' As the ∼3 MK emission is predominantly derived from a smaller volume of AR plasma, the presence of the AR reduces the EM from the quiet Sun values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The average temperatures for all the ARs are determined to be ∼3 MK (blue error bars in Figure 5a), which is close to the “basal” temperature of the AR core reported in earlier research (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Del Zanna and Mason 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Winebarger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The temperature and EM do, however, vary slightly over the course of the AR’s evolution, which is consistent with the observed X- ray light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Following the arrival of AR12749 and AR12758, their activity decayed while rotating on the solar disk (Figure 1), which is why the temperature and EM decreased during their evolution, as indicated by the dashed vertical lines in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' After October 6, 2019, the EM for AR12749 begins to rise as the AR weakens and the quiet Sun emission takes precedence over the AR emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Thus, after the AR has almost died and is very faint, the EM and temperature reach values close to the quiet Sun temperature and EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The temperature and EM for the AR12759 remain almost constant with time, as this AR crossed the solar disk without much decay in activity (Figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Abundance evolution In contrast to the temperature and EM, the abun- dances of Mg, Al, and Si do not follow the X-ray light curve of any of the three ARs throughout their evolu- tion (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The abundances obtained for low-FIP elements Al, Mg, and Si are consistently greater than the photospheric values, demonstrating a persistent FIP bias during the course of the AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' After the emergence of AR12758, the FIP bias is found to be almost constant throughout its decay phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Similarly, during the decay of the AR12749, the FIP bias remains nearly constant, in contrast to certain earlier studies, such as Ko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' They suggested decreasing FIP bias in high- temperature plasma of more than two million degrees during the decay phase of an AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The more established AR, AR12759, which evolved without decaying much during its transit across the solar disk, also shows an almost constant FIP bias, similar to the other two ARs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We do not find any relationship between the age of the AR and the FIP bias, as suggested in some previous papers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=',Del Zanna and Mason 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Doschek and Warren 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The measured abundances for Mg, Si, and S are comparable to those given by (Feldman 1992) and Fludra and Schmelz (1999) (orange shaded regions in Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' However, the Al abundance is ∼30%-60% higher than the coronal abundances reported in the lit- erature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We note that the Al lines in the XSM spectra are blended with Mg lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' From Markov Chain Monte Carlo (MCMC) analysis (discussed in Appendix A), we find that there is no anti-correlation between Mg and Al abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' This suggests that the observed spectra does indeed require higher abundances of Al and cannot be explained by an enhancement of Mg abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' FIP bias at the onset of AR core Though we do not find any relationship between the age of the AR cores and their FIP biases (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='2), 8 5-Mar 17:58 6-Mar 05:58 7-Mar 02:58 7-Mar 12:58 7-Mar 06:58 8-Mar 01:58 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Evolution of the AR12758 during its emergence phase on the solar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Top row shows the evolution of photospheric magnetograms as observed by HMI and bottom row shows the evolution of X-ray emission as observed by XRT Be-thin filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Evolution of the temperature (red points in panel a) and EM (red points in panel b) during the evolution of AR12749, AR12758, and AR12759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' When the ARs were very bright, as bounded by the vertical dashed lines, the magenta bars represent the average values of the temperature and EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The black horizontal dashed lines represent the average temperature and emission measure for the quiet Sun in the absence of any AR reported by Vadawale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The XSM lightcurves of the ARs are shown in grey color, and the lightcurves for the quiescent regions are shown in blue colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' which remain constant, it is interesting to study the timescale on which the FIP bias developed during the emergence of the AR core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Such a study has been made possible using the finer (< one day) time-resolved spec- troscopy during the emerging phase (07-Mar-2020 to 09- Mar-2020) of AR12758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' During this period, we esti- mated the total unsigned photospheric magnetic flux as measured by HMI/SDO and shown in Figure 7a (black color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The peak in the magnetic flux represents the time when the AR completely emerged into the solar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' After the emergence, the unsigned magnetic flux is found to (temporarily) decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Figures 7b and 7c show the evolution of the AR core temperature and emission- measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' With the emergence of the AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The temper- ature becomes close to the AR core temperature of ∼3 AR12749 AR12758 AR12759 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5 (MK) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 b 12 10 8 ６ 4 Sep-30 Oct-03 Oct-06 0ct-09 Mar-06 Mar-10 Mar-14 Mar-28 Apr-01 Apr-05 Apr-09 Date (2019) Date (2020) Date (2020)9 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Panels a-c (red error bars) show the evolution of abundance in the logarithmic scale with A(H)=12 for Mg, Al, and Si during the evolution of AR12749, AR12758 and AR12759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The magenta bars represented the average abundances when the ARs were very bright, as bounded by the vertical dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The y-error bars represent 1σ uncertainty for each parameter, and the x-error bars represent the duration over which a given spectrum is integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The black horizontal dashed lines represent the average abundances for the quiet Sun in the absence of any AR reported by Vadawale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' XSM light curves for each AR are shown in gray in the background, and the blue color on the XSM light curves represents the time duration excluding the flaring activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The range of coronal and photospheric abundances from various authors compiled in the CHIANTI database are shown as orange and green bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The right y-axis shows the FIP bias values for the respective elements with respect to average photospheric abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' MK, and the EM increases as the emitting plasma vol- ume increase until it has emerged completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We also derived the evolution of the FIP bias during this period, shown in Figure 7d for Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' During this period, as the emission from the Mg and Al line complex was weak compared with the background solar emission, the de- rived FIP bias for Mg and Al has a large uncertainty and is not shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Within ∼10 hours of the AR emergence, the FIP bias was already close to 3, and re- mained almost constant throughout the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' So the emerging hot core loops do not show any variation, in agreement with previous suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Recall that the variations in FIP bias reported earlier (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Widing and Feldman 2001) were observed in the cool loops, not the core loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Enhanced bias for Al Figure 8 shows the average values of the FIP bias (relative to the photospheric abundance Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2009)) for all the elements as a function of their FIP values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The lower FIP element, Al (FIP = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='99), is found to have the highest FIP bias of 6-7, whereas the low-FIP elements, Mg (FIP = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='65) and Si (FIP = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='15), are found to have a lower FIP bias of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The mid/high FIP element, S, is found to have a much lower FIP bias of a factor of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A higher FIP bias for Al is note- worthy and may point to an intriguing physical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' However, this may also be a modeling artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' One of the possibilities could be due to missing flux caused by the presence of multi-thermal plasma provid- ing strong signals from emission lines of Al or Mg formed AR12749 AR12758 AR12759 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='2 a 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='6 b 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='2 6 4 bias 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='9 3 N FIP 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='6 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='3 c 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 3 + S 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='4 Sep-30 ( Oct-03 Oct-06 Oct-09 Mar-06 Mar-10 Mar-14 Mar-18Mar-28 Apr-01 Apr-05 Apr-09 Date (2019) Date (2020) Date (2020)10 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Results showing the emerging phase of AR12758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The black curve in panel a shows the evolution of the total unsigned photospheric magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Panel b and c show the evolution of temperature and EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Panel d shows the evolu- tion of FIP bias for Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The dashed lines in panels b-d repre- sent the corresponding parameter for the background solar emission from the rest of the solar-disk except AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The back- ground grey curves in each panel represent the X-ray light curve observed by XSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Whereas the blue curves represent the selected times excluding the flaring period, representing the quiescent AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' To verify this we have simu- lated the emission lines in the energy range of the Mg/Al line complex by considering the isothermal model and a multi-thermal model using the AR DEM of AR12759, reported by Del Zanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022) (see Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='1 in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Similar line intensities from various ion- ization stages of Al and Mg can be seen in both the isothermal and multi-thermal models, confirming that the absence of the flux is not the result of multi-thermal plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Another possibility is that missing flux is caused by missing lines of Al or Mg (mostly satellite lines) that are not yet present in CHIANTI version 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We have analysed the high-resolution spectroscopic observations described by Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1974) and found several ob- served lines that are missing in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' However, the total missing flux, compared to the predicted flux by CHIANTI is not enough to explain the anomalous Al abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' However, the Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1974) obser- vations were taken during a high level of solar activity, so it is possible that the missing lines have a stronger contribution at 3 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Al abundance is currently clearly overestimated by some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Although this analysis is not conclusive enough to rule out Al’s high FIP bias as an artifact, it is also not suf- ficient to conclude that it is not real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A higher Al FIP bias could be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' This might be explained by examin- ing a few particular scenarios from the Ponderomotive force model (Laming 2015) proposed by Laming (pri- vate communication), which could be investigated in a subsequent study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We have also compared the AR core FIP bias obtained with that of the different solar activity levels measured by the XSM in previous research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' These are overplot- ted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The blue points show the FIP bias during the quiet Sun period, which is dominated by X- ray Bright Points (XBP), as reported by Vadawale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' While the green points depict the FIP bias dur- ing the peak of the solar flares as reported by Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The FIP bias of the AR core (red points) shows a consistently higher value for the elements Al, Mg, and Si compared with the FIP bias of XBPs (green points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Since ARs have substantially higher magnetic activity than the XBPs, the increased FIP bias of the ARs relative to the XBPs is expected from the Pondero- motive force model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' On the other hand, chromospheric evaporation during the flaring mechanism results in a near unit FIP bias during the peak of the flares (Mon- dal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' SUMMARY We present the evolution of plasma characteristics for three ARs using disk-integrated soft X-ray spectroscopic observations from the XSM to make simultaneous line and continuum measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Carrying out a compre- hensive study of an AR using the Sun-as-a-star mode observation is challenging because of the presence of multiple activities throughout the solar cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Unique a Mx) 20 (×1021 10 B-flux b 4 (MK) 3 C 10 5 EM 0 d 6 4 bias FIP 2 20 0 20 40 Hours from 07-Mar-202011 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Best fitted parameters for the average spectrum of each AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' AR T EM Mg Al Si S (MK) (1046 cm−3) 12749 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='46+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='24 −0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='05 12759 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='22+0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='32+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='06 6 7 8 9 10 11 FIP (eV) 2 0 2 4 6 8 FIP bias Al Mg Si S AR XBP Flare Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Variation of the FIP bias with the FIP of the ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The red points are the averaged FIP bias for the ARs reported in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The blue points are the FIP bias for the XBPs as reported by Vadawale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The green points are the measured FIP bias during the peak of solar flares as reported by Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' XSM observations made during the minimum of Solar Cycle 24 allowed the study of the evolution of temper- ature, EM, and the abundances of Mg, Al, and Si for the individual ARs in the absence of any other notewor- thy activity on the solar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Since the ARs were the principal contributors of disk-integrated X-rays during their evolution, the temperature and EM followed their X-ray light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The average temperature of all the AR is ∼3 MK, close to the well-known temperature of the AR core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Irrespective of the activity and age of the ARs, the abundances or the FIP biases of Al, Mg, and Si were found to be consistently greater than their photo- spheric values without much variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The abundance values develop within ∼10 hours of the appearance of the AR during its emerging phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Throughout the AR evolution, the low FIP elements, Mg and Si, have a FIP bias close to 3, whereas the mid-FIP element, S, has an average FIP bias of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The lowest FIP element, Al, has a greater FIP bias of ∼6-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' After discussing vari- ous modeling artifacts, the Al abundance appears to be overestimated, although the exact factor is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Increased Al abundance could be real, implying that low-FIP elements degree of FIP bias is linked to their FIP values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Future spectroscopic studies to measure the FIP bias for more low-FIP elements (for example, Ca, whose FIP bias is between Al and Mg) would help us to better understand this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In this regard, recent and upcoming X-ray spectrometers (for example, DAXSS: (Schwab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2020) onboard INSPIRESat-1, SoLEXS (Sankarasubramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2011) onboard up- coming Aditya-L1 observatory, and rocket-borne spec- trometer MaGIXS (Champey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 2022)) will be use- ful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge the use of data from the Solar X- ray Monitor (XSM) on board the Chandrayaan-2 mis- sion of the Indian Space Research Organisation (ISRO), archived at the Indian Space Science Data Centre (ISSDC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The XSM was developed by the engineer- ing team of Physical Research Laboratory (PRL) lead by Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Shanmugam, with support from various ISRO centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We thank various facilities and the technical teams from all contributing institutes and Chandrayaan-2 project, mission operations, and ground segment teams for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Research at PRL is supported by the Department of Space, Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We acknowledge the support from Royal So- ciety through the international exchanges grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' IES\\R2\\170199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' GDZ and HEM acknowledge support from STFC (UK) via the consolidated grant to the atomic astrophysics group at DAMTP, University of Cambridge (ST\\T000481\\1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' AB was the J C Bose Na- tional Fellow during the period of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Martin Laming for the useful discussion on anoma- lous Al abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' APPENDIX 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' RESULTS OF MCMC ANALYSIS We carried out Markov Chain Monte Carlo (MCMC) analysis of the spectra to obtain the regions of parameter space that best fits the observed spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' This was done using the ‘chain’ method available within XSPEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Figure A1 shows the corner plot of the results for the spectrum on 01-Oct-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The results show that all parameters are well constrained by the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Particularly, we note that there is no anti-correlation observed between Al and Mg abundances showing that the enhances Al abundances obtained cannot be adjusted by enhancements in Mg abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Similar trends are observed for spectra of other days as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='85 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='90 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='95 Mg 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='3 Al 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='92 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='95 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='98 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='01 Si 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='36 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='54 T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5 EM 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='85 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='90 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='95 Mg 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='3 Al 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='92 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='95 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='98 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='01 Si 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='5 EM Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Corner plot depicting the results of MCMC analysis for the fitted spectrum on 01-Oct-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The histograms depict the marginalized distribution associated with each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The scatter-plots are overlaid with contours representing 1σ, 2σ, and 3σ levels to show correlations between all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The best-fit parameters are represented by green lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' SIMULATED SPECTRUM To check the effect of temperatures on the Mg/Al line fluxes in the XSM energy range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='55 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='70 keV, we have compared the simulated spectra in the same energy range by considering the isothermal and multi-thermal DEM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='1 shows the simulated 3 MK spectrum (blue) overplotted with the multithermal spectrum (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The isothermal spectrum is generated for an emission measure of 1027 cm−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The multithermal spectrum is derived by 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='70 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='0 Normalized intensity Mg XI, Al XI-XII Al XII Mg XI Mg XI DEM Isothermal Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Simulated spectra from CHIANTI v 10 in the energy range of Mg/Al line complex of XSM observed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Solid blue curve show the multi-thermal spectrum and dashed orange curve shows the isothermal spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' using the reported quiescent AR DEM by Del Zanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022), which was obtained from the Hinode EIS observation of AR12759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' For the comparison of both spectra, we have normalized them with the corresponding line flux of Mg XI, and Al XI-XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Similar line intensities predicted by both isothermal and multithermal models indicates that spectra are insensitive to temperature in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 14 REFERENCES Arnaud, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Dorman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Gordon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' XSPEC: An X-ray spectral fitting package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Asplund, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Grevesse, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Sauval, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Scott, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Chemical Composition of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ARA&A, 47(1):481–522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Baker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', D´emoulin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', van Driel-Gesztelyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Green, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Steed, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Carlyle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Plasma Composition in a Sigmoidal Anemone Active Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 778(1):69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Baker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', D´emoulin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Yardley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', van Driel-Gesztelyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Long, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Green, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' FIP Bias Evolution in a Decaying Active Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 802(2):104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Baker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', van Driel-Gesztelyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', James, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', D´emoulin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Long, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Warren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Williams, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Coronal Elemental Abundances in Solar Emerging Flux Regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 856(1):71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Bochsler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Minor ions in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A&A Rv, 14(1):1–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Baker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', van Driel-Gesztelyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Warren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A solar cycle correlation of coronal element abundances in sun-as-a-star observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Nature Communications, 8(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Ugarte-Urra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Warren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Full-Sun observations for identifying the source of the slow solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Nature Communications, 6:5947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Warren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Establishing a Connection Between Active Region Outflows and the Solar Wind: Abundance Measurements with EIS/Hinode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJL, 727(1):L13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Champey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Winebarger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Kobayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Athiray, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Hertz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Savage, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Beabout, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Beabout, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Broadway, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Bruccoleri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Cheimets, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Davis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Duffy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Golub, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Gregory, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Griffith, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Haight, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Heilmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Hogue, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Hohl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Hyde, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Kegley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Kolodzieczjak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Ramsey, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Ranganathan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Robertson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Schattenburg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Speegle, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Vigil, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Walsh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Weddenorf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Wright, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Marshall Grazing Incidence X-ray Spectrometer (MaGIXS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Journal of Astronomical Instrumentation, 11(2):2250010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Dahlburg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Laming, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Taylor, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Obenschain, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' PONDEROMOTIVE ACCELERATION IN CORONAL LOOPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Astrophysical Journal, 831(2):160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Solar active regions: The footpoints of 1 MK loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A&A, 406:L5–L8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Benchmarking atomic data for the CHIANTI atomic database: coronal lines observed by Hinode EIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A&A, 537:A38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The multi-thermal emission in solar active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A&A, 558:A73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The EUV spectrum of the Sun: Quiet- and active-Sun irradiances and chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A&A, 624:A36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Mason, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Solar active regions: SOHO/CDS and TRACE observations of quiescent coronal loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A&A, 406:1089–1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Mason, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Elemental abundances and temperatures of quiescent solar active region cores from X-ray observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A&A, 565:A14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Mason, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Solar UV and X-ray spectral diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Living Reviews in Solar Physics, 15(1):5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Del Zanna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mondal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Rao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mithun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Vadawale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Reeves, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mason, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Sarkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Janardhan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Bhardwaj, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Multiwavelength Observations by XSM, Hinode, and SDO of an Active Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Chemical Abundances and Temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 934(2):159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Doschek, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Warren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Variability of Solar Coronal Abundances in Active Regions and the Quiet Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 884(2):158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Dwivedi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Curdt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Wilhelm, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Analysis of Extreme-Ultraviolet Off-Limb Spectra Obtained with SUMER/SOHO: Ne VI-Mg VI Emission Lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 517(1):516–525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Feldman, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Elemental abundances in the upper solar atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' PhyS, 46(3):202–220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Feldman, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Landi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Doschek, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Dammasch, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Curdt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Free-Free Emission in the Far-Ultraviolet Spectral Range: A Resource for Diagnosing Solar and Stellar Flare Plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 593(2):1226–1241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Feldman, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Schuhle, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Widing, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Laming, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Coronal composition above the solar equator and the north pole as determined from spectra acquired by the SUMER instrument onSOHO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Astrophysical Journal, 505(2):999–1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Feldman, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Widing, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A review of the first ionization potential effect on elemental abundances in the solar corona and in flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Physics of Plasmas, 9(2):629–635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Fludra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Schmelz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The absolute coronal abundances of sulfur, calcium, and iron from Yohkoh-BCS flare spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A&A, 348:286–294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 15 Freeland, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Handy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Data Analysis with the SolarSoft System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' SoPh, 182(2):497–500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Gloeckler, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Geiss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The abundances of elements and isotopes in the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In Waddington, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', editor, Cosmic Abundances of Matter, volume 183 of American Institute of Physics Conference Series, pages 49–71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Ko, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Young, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Muglach, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Warren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Ugarte-Urra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Correlation of Coronal Plasma Properties and Solar Magnetic Field in a Decaying Active Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 826(2):126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Lakshitha, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mondal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Narendranath, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Paul, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Elemental abundances during A-class solar flares: Soft X-ray spectroscopy from Chandrayaan-2 XSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Under preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Laming, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A Unified Picture of the First Ionization Potential and Inverse First Ionization Potential Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 614(2):1063–1072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Laming, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Non-Wkb Models of the First Ionization Potential Effect: Implications for Solar Coronal Heating and the Coronal Helium and Neon Abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 695(2):954–969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Laming, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Non-WKB Models of the First Ionization Potential Effect: The Role of Slow Mode Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 744(2):115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Laming, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The fip and inverse fip effects in solar and stellar coronae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Living Reviews in Solar Physics, 12:1–76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Laming, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The first ionization potential effect from the ponderomotive force: On the polarization and coronal origin of alfv´en waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Astrophysical Journal, 844(2):153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Mihailescu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Baker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Green, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', van Driel-Gesztelyi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Long, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and To, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' What Determines Active Region Coronal Plasma Composition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 933(2):245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Mithun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Vadawale, S.' metadata={'source': 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Sarkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Kumar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Jangid, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Sarda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Shah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Bhardwaj, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Data processing software for chandrayaan-2 solar x-ray monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Astronomy and Computing, 34:100449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Mithun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Vadawale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Sarkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Shanmugam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Patel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mondal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Joshi, B.' 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H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Bhardwaj, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Solar X-Ray Monitor on Board the Chandrayaan-2 Orbiter: In-Flight Performance and Science Prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' SoPh, 295(10):139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Mithun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Vadawale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Zanna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Rao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Submitted to ApJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Mondal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Sarkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Vadawale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mithun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Janardhan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Del Zanna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mason, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mitra-Kraev, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Narendranath, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Evolution of Elemental Abundances during B-Class Solar Flares: Soft X-Ray Spectral Measurements with Chandrayaan-2 XSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 920(1):4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Pottasch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Lower Solar Corona: Interpretation of the Ultraviolet Spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 137:945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Saba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Strong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Coronal abundances of O, Ne, Mg, and Fe in solar active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Advances in Space Research, 13(9):391–394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Sankarasubramanian, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Ramadevi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Bug, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Umapathy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Seetha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Sreekumar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Kumar (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' SoLEXS - A low energy X-ray spectrometer for solar coronal studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' In Astronomical Society of India Conference Series, volume 2 of Astronomical Society of India Conference Series, pages 63–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Schwab, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Sewell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Woods, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Caspi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mason, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Moore, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Soft X-Ray Observations of Quiescent Solar Active Regions Using the Novel Dual-zone Aperture X-Ray Solar Spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 904(1):20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Shanmugam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Vadawale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Patel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Adalaja, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mithun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Ladiya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Goyal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Tiwari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Singh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Painkra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Acharya, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Bhardwaj, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Hait, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Patinge, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Kapoor, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Kumar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Satya, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Saxena, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Arvind, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Solar X-ray Monitor Onboard Chandrayaan-2 Orbiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Current Science, 118(1):45–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Sheeley, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' A Volcanic Origin for High-FIP Material in the Solar Atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 440:884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Sheeley, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Elemental Abundance Variations in the Solar Atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 469:423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Testa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Martinez-Sykora, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and De Pontieu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Coronal Abundances in an Active Region: Evolution and Underlying Chromospheric and Transition Region Properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' arXiv e-prints, page arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content='07755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' 16 Vadawale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Shanmugam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Acharya, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Patel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Goyal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Shah, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Hait, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Patinge, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Subrahmanyam, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Solar x-ray monitor (xsm) on-board chandrayaan-2 orbiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Advances in Space Research, 54(10):2021 – 2028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Lunar Science and Exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Vadawale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mithun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mondal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Sarkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Janardhan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Joshi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Bhardwaj, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Shanmugam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Patel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Adalja, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Goyal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Ladiya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Tiwari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Singh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Observations of the quiet sun during the deepest solar minimum of the past century with chandrayaan-2 XSM: Sub-a-class microflares outside active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Astrophysical Journal Letters, 912(1):L13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Vadawale, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mondal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mithun, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Sarkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Janardhan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Joshi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Bhardwaj, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Shanmugam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Patel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Adalja, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Goyal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Ladiya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Tiwari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Singh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Observations of the quiet sun during the deepest solar minimum of the past century with chandrayaan-2 XSM: Elemental abundances in the quiescent corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Astrophysical Journal Letters, 912(1):L12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Walker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Rugge, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Weiss, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Relative Coronal Abundances Derived from X-Ray Observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Sodium, Magnesium, Aluminum, Silicon, Sulfur, and Argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 188:423–440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Widing, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Emerging Active Regions on the Sun and the Photospheric Abundance of Neon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 480(1):400–405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Widing, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Feldman, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Nonphotospheric abundances in a solar active region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 416:392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Widing, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Feldman, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' On the Rate of Abundance Modifications versus Time in Active Region Plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJ, 555(1):426–434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Winebarger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Warren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Schmelz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Cirtain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Mulu-Moore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', Golub, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=', and Kobayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Defining the “Blind Spot” of Hinode EIS and XRT Temperature Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' ApJL, 746(2):L17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' Young, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' and Mason, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' The Mg/Ne abundance ratio in a recently emerged flux region observed by CDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} +page_content=' SoPh, 175(2):523–539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE1T4oBgHgl3EQf6AXQ/content/2301.03519v1.pdf'} diff --git a/4dE1T4oBgHgl3EQf6QUq/content/tmp_files/2301.03520v1.pdf.txt b/4dE1T4oBgHgl3EQf6QUq/content/tmp_files/2301.03520v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..64fe51699d27979d22ada47d3a650c2a1528040b --- /dev/null +++ b/4dE1T4oBgHgl3EQf6QUq/content/tmp_files/2301.03520v1.pdf.txt @@ -0,0 +1,715 @@ +arXiv:2301.03520v1 [math.FA] 9 Jan 2023 +CLASSIFYING WEAK PHASE RETRIEVAL +P. G. CASAZZA AND F. AKRAMI +Abstract. We will give several surprising equivalences and consequences of +weak phase retrieval. These results give a complete understanding of the dif- +ference between weak phase retrieval and phase retrieval. We also answer two +longstanding open problems on weak phase retrieval: (1) We show that the +families of weak phase retrievable frames {xi}m +i=1 in Rn are not dense in the +family of m-element sets of vectors in Rn for all m ≥ 2n − 2; (2) We show +that any frame {xi}2n−2 +i=1 +containing one or more canonical basis vectors in Rn +cannot do weak phase retrieval. We provide numerous examples to show that +the obtained results are best possible. +1. Introduction +The concept of frames in a separable Hilbert space was originally introduced by +Duffin and Schaeffer in the context of non-harmonic Fourier series [14]. Frames +are a more flexible tool than bases because of the redundancy property that make +them more applicable than bases. Phase retrieval is an old problem of recovering +a signal from the absolute value of linear measurement coefficients called intensity +measurements. Phase retrieval and norm retrieval have become very active areas of +research in applied mathematics, computer science, engineering, and more today. +Phase retrieval has been defined for both vectors and subspaces (projections) in all +separable Hilbert spaces, (e.g., [3], [4], [5], [6], [9], [10] and [11]). +The concept of weak phase retrieval weakened the notion of phase retrieval and it +has been first defined for vectors in ([8] and [7]). The rest of the paper is organized +as follows: In Section 2, we give the basic definitions and certain preliminary results +to be used in the paper. Weak phase retrieval by vectors is introduced in section +3. In section 4 we show that any family of vectors {xi}2n−2 +i=1 +doing weak phase +retrieval cannot contain a unit vector. In section 5, we show that the weak phase +retrievable frames are not dense in all frames. And in section 6 we give several +surprising equivalences and consequences of weak phase retrieval. These results +give a complete understanding of the difference between weak phase retrieval and +phase retrieval. +2. preliminaries +First we give the background material needed for the paper. Let H be a finite +or infinite dimensional real Hilbert space and B(H) the class of all bounded linear +operators defined on H. The natural numbers and real numbers are denoted by +“N” and “R”, respectively. We use [m] instead of the set {1, 2, 3, . . ., m} and use +[{xi}i∈I] instead of span{xi}i∈I, where I is a finite or countable subset of N. We +2010 Mathematics Subject Classification. 42C15, 42C40. +Key words and phrases. Real Hilbert frames, Full spark, Phase retrieval, Weak phase retrieval. +The first author was supported by NSF DMS 1609760. +1 + +2 +P. G. CASAZZA AND F. AKRAMI +denote by Rn a n dimensional real Hilbert space. We start with the definition of a +real Hilbert space frame. +Definition 1. A family of vectors {xi}i∈I in a finite or infinite dimensional separable +real Hilbert space H is a frame if there are constants 0 < A ≤ B < ∞ so that +A∥x∥2 ≤ +� +i∈I +|⟨x, xi⟩|2 ≤ B∥x∥2, +for all +f ∈ H. +The constants A and B are called the lower and upper frame bounds for {xi}i∈I, +respectively. If an upper frame bound exists, then {xi}i∈I is called a B-Bessel +seqiemce or simply Bessel when the constant is implicit. If A = B, it is called an +A-tight frame and in case A = B = 1, it is called a Parseval frame. The values +{⟨x, xi⟩}∞ +i=1 are called the frame coefficients of the vector x ∈ H. +It is immediate that a frame must span the space. We will need to work with +Riesz sequences. +Definition 2. A family X = {xi}i∈I in a finite or infinite dimensional real Hilbert +space H is a Riesz sequence if there are constants 0 < A ≤ B < ∞ satisfying +A +� +i∈I +|ci|2 ≤ ∥ +� +i∈I +cixi∥2 ≤ B +� +i∈I +|ci|2 +for all sequences of scalars {ci}i∈I. If it is complete in H, we call X a Riesz basis. +For an introduction to frame theory we recommend [12, 13]. +Throughout the paper the orthogonal projection or simply projection will be a self- +adjoint positive projection and {ei}∞ +i=1 will be used to denote the canonical basis +for the real space Rn, i.e., a basis for which +⟨ei, ej⟩ = δi,j = +� +1 +if i = j, +0 +if i ̸= j. +Definition 3. A family of vectors {xi}i∈I in a real Hilbert space H does phase +(norm) retrieval if whenever x, y ∈ H, satisfy +|⟨x, xi⟩| = |⟨y, xi⟩| +for all i ∈ I, +then x = ±y +(∥x∥ = ∥y∥). +Phase retrieval was introduced in reference [4]. See reference [1] for an introduc- +tion to norm retrieval. +Note that if {xi}i∈I does phase (norm) retrieval, then so does {aixi}i∈I for any +0 < ai < ∞ for all i ∈ I. But in the case where |I| = ∞, we have to be careful to +maintain frame bounds. This always works if 0 < infi∈I ai ≤ supi∈Iai < ∞. But +this is not necessary in general [1]. The complement property is an essential issue +here. +Definition 4. A family of vectors {xi}i∈I in a finite or infinite dimensional real +Hilbert space H has the complement property if for any subset J ⊂ I, +either span{xi}i∈J = H +or +span{xi}i∈Jc = H. +Fundamental to this area is the following for which the finite dimensional case +appeared in [10]. + +WEAK PHASE RETRIEVAL +3 +Theorem 1. A family of vectors {xi}i∈I does phase retrieval in Rn if and only if it +has the complement property. +We recall: +Definition 5. A family of vectors {xi}m +i=1 in Rn is full spark if for every I ⊂ +[m] with |I| = n , {xi}i∈I is linearly independent. +Corollary 1. If {xi}m +i=1 does phase retrieval in Rn, then m ≥ 2n− 1. If m = 2n− 1, +{xi}m +i=1 does phase retrieval if and only if it is full spark. +We rely heavily on a significant result from [2]: +Theorem 2. If {xi}2n−2 +i=1 +does weak phase retrieval in Rn then for every I ⊂ [2n−2], +if x ⊥ span{xi}i∈I and y ⊥ {xi}i∈Ic then +x +∥x∥ + +y +∥y∥ and +x +∥x∥ − +y +∥y∥ are disjointly +supported. In particular, if ∥x∥ = ∥y∥ = 1, then x + y and x − y are disjointly +supported. Hence, if x = (a1, a2, . . . , an) then y = (ǫ1a1, ǫ2a2, . . . , ǫnan), where +ǫi = ±1 for i = 1, 2, . . ., n. +Remark 2.1. The above theorem may fail if ∥x∥ ̸= ∥y∥. For example, consider the +weak phase retrievable frame in R3: + + +1 +1 +1 +−1 +1 +1 +1 +−1 +1 +1 +1 +−1 + + +Also, x = (0, 1, −1) is perpendicular to rows 1 and 2 and y = (0, 1 +2, 1 +2) is orthogonal +to rows 2 and 3. But x + y = (0, 3 +2, 1 +2) and x − y = (0, −1 +2 , −3 +2 ) and these are not +disjointly supported. But if we let them have the same norm we get x = (0, 1, −1) +and y = (0, 1, 1) so x + y = (0, 1, 0) and x − y = (0, 0, 1) and these are disjointly +supported. +3. Weak phase retrieval +The notion of “Weak phase retrieval by vectors” in Rn was introduced in [8] and +was developed further in [7]. One limitation of current methods used for retrieving +the phase of a signal is computing power. Recall that a generic family of (2n − 1)- +vectors in Rn satisfies phaseless reconstruction, however no set of (2n − 2)-vectors +can (See [7] for details). By generic we are referring to an open dense set in the set +of (2n − 1)-element frames in Rn. +Definition 6. Two vectors x = (a1, a2, . . . , an) and y = (b1, b2, . . . , bn) in Rn weakly +have the same phase if there is a |θ| = 1 so that phase(ai) = θphase(bi) for all +i ∈ [n], for which ai ̸= 0 ̸= bi. +If θ = 1, we say x and y weakly have the same signs and if θ = −1, they weakly +have the opposite signs. +Therefore with above definition the zero vector in Rn weakly has the same phase +with all vectors in Rn. For x ∈ R, sgn(x) = 1 if x > 0 and sgn(x) = −1 if x < 0. +Definition 7. A family of vectors {xi}m +i=1 does weak phase retrieval in Rn if for +any x = (a1, a2, . . . , an) and y = (b1, b2, . . . , bn) in Rn with |⟨x, xi⟩| = |⟨y, xi⟩| for +all i ∈ [m], then x and y weakly have the same phase. +A fundamental result here is + +4 +P. G. CASAZZA AND F. AKRAMI +Proposition 1. [8] Let x = (a1, a2, . . . , an) and y = (b1, b2, . . . , bn) in Rn. +The +following are equivalent: +(1) We have sgn(aiaj) = sgn(bibj), for all 1 ≤ i ̸= j ≤ n +(2) Either x, y have weakly the same sign or they have the opposite signs. +It is clear that if {xi}m +i=1 does weak phase retrieval in Rn, then {cixi}m +i=1 does +weak phase retrieval as long as ci > 0 for all i = 1, 2, . . ., m. +The following appears in [7]. +Theorem 3. If X = {xi}m +i=1 does weak phase retrieval in Rn, then m ≥ 2n − 2. +Finally, we have: +Theorem 4. [7] If a frame X = {xi}2n−2 +i=1 +does weak phase retrieval in Rn, then X +is a full spark frame. +Clearly the converse of above theorem is not hold, for example {(1, 0), (0, 1)} is +full spark frame that fails weak phase retrieval in R2. +If {xi}i∈I does phase retrieval and R is an invertible operator on the space +then {Rxi}i∈I does phase retrieval. This follows easily since |⟨x, Rxi⟩| = |⟨y, Rxi⟩| +implies |⟨R∗x, xi⟩| = |⟨R∗y, xi⟩|, and so R∗x = θR∗y for |θ| = 1. +Since R is +invertible, x = θy. This result fails badly for weak phase retrieval. For example, +let e1 = (1, 0), e2 = (0, 1), x1 = ( 1 +√ +2, +1 +√ +2, x2 = ( 1 +√ +2, −1 +√ +2) in R2. Then {e1, e2} fails +weak phase retrieval, {x1, x2} does weak phase retrieval and Uei = xi is a unitary +operator. +4. Frames Containing Unit Vectors +Theorem 5. Any frame {xi}2n−2 +i=1 +whith one or more canonical basis vectors in Rn +cannot do weak phase retrieval. +Proof. We proceed by way of contradiction. Recall that {xi}2n−2 +i=1 +must be full spark. +Let {ei}n +i=1 be the canonical orthonormal basis of Rn. Assume I ⊂ {1, 2, . . ., 2n−2} +with |I| = n − 1 and assume x = (a1, a2, . . . , an), y = (b1, b2, . . . , bn) with ∥x∥ = +∥y∥ = 1 and x ⊥ X = span{xi}i∈I and y ⊥ span{xi}2n−2 +i=n . After reindexing {ei}n +i=1 +and {xi}2n−2 +i=1 }, we assume x1 = e1, I = {1, 2, . . ., n−1 and Ic = {n, n+1, . . . , 2n− +2}. Since ⟨x, x1⟩ = a1 = 0, by Theorem 2, b1 = 0. Let P be the projection on +span{ei}n +i=2. So {Pxi}2n−2 +i=n +is (n − 1)-vectors in an (n − 1)-dimensional space and +y is orthogonal to all these vectors. So there exist {ci}2n−2 +i=n +not all zero so that +2n−2 +� +i=n +ciPxi = 0 and so +2n−1 +� +i=n +cixi(1)x1 − +2n−2 +� +i=n +cixi = 0. +That is, our vectors are not full spark, a contradiction. +□ +Remark 4.1. The fact that there are (2n− 2) vectors in the theorem is critical. For +example, e1, e2, e1 + e2 is full spark in R2, so it does phase retrieval - and hence +weak phase retrieval - despite the fact that it contains both basis vectors. +The converse of Theorem 5 is not true in general. +Example 1. Consider the full spark frame X = {(1, 2, 3), (0, 1, 0), (0, −2, 3), (1, −2, −3)} +in R3. Every set of its two same coordinates, +{(1, 2), (0, 1), (0, −2), (1, −2)}, {(1, 3), (0, 0), (0, 3), (1, −3)}, and + +WEAK PHASE RETRIEVAL +5 +{(2, 3), (1, 0), (−2, 3), (−2, −3)} +do weak phase retrieval in R2, but by Theorem 5, X cannot do weak phase retrieval +in R3. +5. Weak Phase Retrievable Frames are not Dense in all Frames +If m ≥ 2n − 1 and {xi}m +i=1 is full spark then it has complement property and +hence does phase retrieval. Since the full spark frames are dense in all frames, it +follows that the frames doing phase retrieval are dense in all frames with ≥ 2n − 1 +vectors. We will now show that this result fails for weak phase retrievable frames. +The easiest way to get very general frames failing weak phase retrieval is: +Choose x, y ∈ Rn so that x + y, x − y do not have the same or opposite signs. +Let X1 = x⊥ and Y1 = y⊥. Then span{X1, X2} = Rn. Choose {xi}n−1 +i=1 vectors +spanning X1 and {xi}2n−2 +i=n +be vectors spanning X2. Then {xi}2n−2 +i=1 +is a frame for +Rn with x ⊥ xi, for i = 1, 2, . . ., n − 1 and y ⊥ xi, for all i = n, n + 1, , . . . , 2n − 2. +It follows that +|⟨x + y, xi⟩| = |⟨x − y, xi⟩|, for all i = 1, 2, . . . , n, +but x, y do not have the same or opposite signs and so {xi}2n−2 +i=1 +fails weak phase +retrieval. +Definition 8. If X is a subspace of Rn, we define the sphere of X as +SX = {x ∈ X : ∥x∥ = 1}. +Definition 9. If X, Y are subspaces of Rn, we define the distance between X +and Y as +d(X, Y ) = supx∈SXinfy∈SY ∥x − y∥. +It follows that if d(X, Y ) < ǫ then for any x ∈ X there is a z ∈ SY so that +∥ x +∥x∥ − z∥ < ǫ. Letting y = ∥x∥z we have that ∥y∥ = ∥x∥ and ∥x − y∥ < ǫ∥x∥. +Proposition 2. Let X, Y be hyperplanes in Rn and unit vectors x ⊥ X, y ⊥ Y . If +d(X, Y ) < ǫ then min{∥x − y∥, ∥x + y∥} < 6ǫ. +Proof. Since span{y, Y } = Rn, x = ay + z for some z ∈ Y . By replacing y by −y +if necessary, we may assume 0 < a. By assumption, there is some w ∈ X with +∥w∥ = ∥z∥ so that ∥w − z∥ < ǫ. Now +a = a∥y∥ = ∥ay∥ = ∥x − z∥ ≥ ∥x − w∥ − ∥w − z∥ ≥ ∥x∥ − ǫ = 1 − ǫ. +So, 1 − a < ǫ. Also, 1 = ∥x∥2 = a2 + ∥w∥2 implies a < 1. I.e. 0 < 1 − a < ǫ. +1 = ∥x∥2 = ∥ay + z∥2 = a2∥y∥2 + ∥z∥2 = a2 + ∥z∥2 ≥ (1 − ǫ)2 + ∥z∥2. +So +∥z∥2 ≤ 1 − (1 − ǫ)2 = 2ǫ − ǫ2 ≤ 2ǫ. +Finally, +∥x − y∥2 = ∥(ay + z) − y∥2 +≤ (∥(1 − a)y∥ + ∥z∥)2 +≤ (1 − a)2∥y∥2 + ∥z∥2 + 2(1 − a)∥y∥∥z∥ +< ǫ2 + 2ǫ + 2 +√ +2ǫ2 +< 6ǫ. + +6 +P. G. CASAZZA AND F. AKRAMI +□ +Lemma 1. Let X, Y be hyperplanes in Rn, {xi}n−1 +i=1 be a unit norm basis for X and +{yi}n−1 +i=1 be a unit norm basis for Y with basis bounds B. If �n−1 +i=1 ∥xi − yi∥ < ǫ +then d(X, Y ) < 2ǫB. +Proof. Let 0 < A ≤ B < ∞ be upper and lower basis bounds for the two bases. +Given a unit vector x = �n−1 +i=1 aixi ∈ X, let y = �n−1 +i=1 aiyi ∈ Y . We have that +sup1≤i≤n−1|ai| ≤ B. We compute: +∥x − y∥ = ∥ +n−1 +� +i=1 +ai(xi − yi)∥ +≤ +n−1 +� +i=1 +|ai|∥xi − yi∥ +≤ (sup1≤i≤n−1|ai|) +n−1 +� +i=1 +∥xi − yi∥ ≤ Bǫ. +So +∥y∥ ≥ ∥x∥ − ∥x − y∥ ≥ 1 − Bǫ. +����x − +y +∥y∥ +���� ≤ ∥x − y∥ + +����y − +y +∥y∥ +���� +≤ Bǫ + +1 +∥y∥∥(1 − ∥y∥)y∥ += Bǫ + (1 − ∥y∥) +≤ 2Bǫ. +It follows that d(X, Y ) < 2Bǫ. +□ +Lemma 2. Let {xi}n +i=1 be a basis for Rn with unconditional basis constant B and +assume yi ∈ Rn satisfies �n +i=1 ∥xi − yi∥ < ǫ. Then {yi}n +i=1 is a basis for Rn which +is 1 + ǫB-equivalent to {xi}n +i=1 and has unconditional basis constant B(1 + ǫB)2. +Proof. Fix {ai}n +i=1 and compute +∥ +n +� +i=1 +aiyi∥ ≤ ∥ +n +� +i=1 +aixi∥ + ∥ +n +� +i=1 +|ai|(xi − yi)∥ +≤ ∥ +n +� +i=1 +aixi∥ + (sup1≤i≤n|ai|) +n +� +i=1 +∥xi − yi∥ +≤ ∥ +n +� +i=1 +aixi∥ + (sup1≤i≤n|ai|)ǫ +≤ ∥ +n +� +i=1 +aixi∥ + ǫB∥ +n +� +i=1 +aixi∥ += (1 + ǫB)∥ +n +� +i=1 +aixi∥. + +WEAK PHASE RETRIEVAL +7 +Similarly, +∥ +n +� +i=1 +|ai|yi∥ ≥ (1 − ǫB)∥ +n +� +i=1 +aixi∥. +So {xi}n +i=1 is (1 + ǫB)-equivalent to {yi}n +i=1. +For ǫi = ±1, +∥ +n +� +i=1 +ǫiaiyi∥ ≤ (1 + ǫB)∥ +n +� +i=1 +ǫiaixi∥ +≤ B(1 + ǫB)∥ +n +� +i=1 +aixi∥ +≤ B(1 + ǫB)2∥ +n +� +i=1 +aiyi∥. +and so {yi}n +i=1 is a B(1 + ǫB) unconditional basis. +□ +Theorem 6. The family of m-element weak phase retrieval frames are not dense in +the set of m-element frames in Rn for all m ≥ 2n − 2. +Proof. We may assume m = 2n−2 since for larger m we just repeat the (2n-2) vec- +tors over and over until we get m vectors. Let {ei}n +i=1 be the canonical orthonormal +basis for Rn and let xi = ei for i = 1, 2, . . . , n. By [10], there is an orthonormal +sequence {xi}2n−2 +i=n+1 so that {xi}2n−2 +i=1 +is full spark. Let I = {1, 2, . . ., n − 1}. Let +X = span{xi}n−1 +i=1 and Y = span{xi}2n−2 +i=n . +Then x = en ⊥ X and there is a +∥y∥ = 1 with y ⊥ Y . +Note that ⟨x − y, en⟩ ̸= 0 ̸= ⟨x + y, en⟩, for otherwise, +x = ±y ⊥ span{xi}i̸=n, contradicting the fact that the vectors are full spark. So +there is a j = n and a δ > 0 so that |(x + y)(j)|, |(x − y)(j)| ≥ δ. We will show +that there exists an 0 < ǫ so that whenever {yi}2n−2 +i=1 +are vectors in Rn satisfying +�n +i=1 ∥xi − yi∥ < ǫ, then {yi}n +i=1 fails weak phase retrieval. +Fix 0 < ǫ. Assume {yi}2n−2 +i=1 +are vectors so that �2n−2 +i=1 +∥xi−yi∥ < ǫ. Choose unit +vectors x′ ⊥ span{yi}i∈I, y′ ⊥ span{yi}i∈Ic. By Proposition 2 and Lemma 1, we +may choose ǫ small enough (and change signs if necessary) so that ∥x−x′∥, ∥y−y′∥ < +δ +4B . Hence, since the unconditional basis constant is B, +|[(x + y) − (x′ + y′)](j)| +≤ |(x − x′)j| + |(y − y′)(j)| +< B∥x − x′∥ + B∥y − y′∥ +≤ 2B δ +4B = δ +2. +It follows that +|(x′ + y′)(j)| ≥ |(x + y)(j)| − |[(x + y) − (x′ + y′)](j)| ≥ δ − 1 +2δ = δ +2. +Similarly, |(x′ − y′)(j)| > δ +2. So x′ + y′, x′ − y′ are not disjointly supported and so +{yi}2n−2 +i=1 +fails weak phase retrieval by Theorem 2. +□ +6. Classifying Weak Phase Retrieval +In this section we will give several surprising equivalences and consequences of +weak phase retrieval. These results give a complete understanding of the difference +between weak phase retrieval and phase retrieval. +Now we give a surprising and very strong classification of weak phase retrieval. + +8 +P. G. CASAZZA AND F. AKRAMI +Theorem 7. Let {xi}2n−2 +i=1 +be non-zero vectors in Rn. The following are equivalent: +(1) The family {xi}2n−2 +i=1 +does weak phase retrieval in Rn. +(2) If x, y ∈ Rn and +(6.1) +|⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, . . . , 2n − 2, +then one of the following holds: +(a) x = ±y. +(b) x and y are disjointly supported. +Proof. (1) ⇒ (2): Given the assumption in the theorem, assume (a) fails and we will +show that (b) holds. Let x = (a1, a2, . . . , an), y = (b1, b2, . . . , bn). Since {xi}2n−2 +i=1 +does weak phase retrieval, replacing y by −y if necessary, Equation 6.1 implies +aj = bj whenever aj ̸= 0 ̸= bj. +Let +I = {1 ≤ i ≤ 2n − 2 : ⟨x, xi⟩ = ⟨y, yi⟩. +Then +x + y ⊥ xi for all i ∈ Ic and x − y ⊥ xi for all i ∈ I. +By Theorem 2, +x + y +∥x + y + +x − y +∥x − y∥ and +x + y +∥x + y∥ − +x − y +∥x − y∥ are disjointly supported. +Assume there is a 1 ≤ j ≤ n with aj = bj ̸= 0. Then +(x + y)(j) +∥x + y∥ ++ (x − y)(j) +∥x − y∥ += +2aj +∥x + y∥ and (x + y)(j) +∥x + y∥ +− (x − y)(j) +∥x − y∥ += +2aj +∥x + y∥, +Contradicting Theorem 2. +(2) ⇒ (1): This is immediate since (a) and (b) give the conditions for weak phase +retrieval. +□ +Phase retrieval is when (a) in the theorem holds for every x, y ∈ Rn. So this the- +orem shows clearly the difference between weak phase retrieval and phase retrieval: +namely when (b) holds at least once. +Corollary 2. If {xi}2n−2 +i=1 +does weak phase retrieval in Rn, then there are disjointly +supported non-zero vectors x, y ∈ Rn satisfying: +|⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, . . . , 2n − 2. +Proof. Since {xi}2n−2 +i=1 +must fail phase retrieval, (b) of Theorem 7 must hold at least +once. +□ +Definition 10. Let {ei}n +i=1 be the canonical orthonormal basis of Rn. If J ⊂ [n], +we define PJ as the projection onto span{ei}i∈J. +Theorem 8. Let {xi}m +i=1 be unit vectors in Rn. The following are equivalent: +(1) Whenever I ⊂ [2n − 2] and 0 ̸= x ⊥ xi for i ∈ I and 0 ̸= y ⊥ xi for i ∈ Ic, +there is no j ∈ [n] so that ⟨x, ej⟩ = 0 = ⟨y, ej⟩. +(2) For every J ⊂ [n] with |J| = n − 1, {Pjxi}2n−2 +i=1 +does phase retrieval. +(3) For every J ⊂ [n] with |J| < n, {PJxi}2n−2 +i=1 +does phase retrieval. + +WEAK PHASE RETRIEVAL +9 +Proof. (1) ⇒ (2): We prove the contrapositive. So assume (2) fails. Then choose +J ⊂ [n] with |J| = n − 1, J = [n] \ {j}, and {PJxi}2n−2 +i=1 +fails phase retrieval. In +particular, it fails complement property. That is, there exists I ⊂ [2n− 2] and span +{PJxi}i∈I ̸= PJRn and span {Pjxi}i∈Ic ̸= PJRn. So there exists norm one vectors +x, y in PJRn with PJx = x ⊥ PJxi for all i ∈ I and PJy = y ⊥ PJxi for all i ∈ Ic. +Extend x, y to all of Rn by setting x(j) = y(j) = 0. Hence, x ⊥ xi for i ∈ I and +y ⊥ xi for i ∈ Ic, proving (1) fails. +(2) ⇒ (3): This follows from the fact that every projection of a set of vectors +doing phase retrieval onto a subset of the basis also does phase retrieval. +(3) ⇒ (2): This is obvious. +(3) ⇒ (1): We prove the contrapositive. So assume (1) fails. Then there is a +I ⊂ [2n− 2] and 0 ̸= x ⊥ xi for i ∈ I and 0 ̸= y ⊥ xi for i ∈ Ic and a j ∈ [n] so that +⟨x, ej⟩ = ⟨y, ej⟩ = 0. It follows that x = PJx, y = PJy are non zero and x ⊥ Pjxi +for all i ∈ I and y ⊥ Pjxi for i ∈ Ic, so {PJxi}2n−2 +i=1 +fails phase retrieval. +□ +Remark 6.1. The assumptions in the theorem are necessary. That is, in general, +{xi}m +i=1 can do weak phase retrieval and {PJxi}m +i=1 may fail phase retrieval. For +example, in R3 consider the row vectors {xi}4 +i=1 of: + + +1 +1 +1 +−1 +1 +1 +1 +−1 +1 +1 +1 +−1 + + +This set does weak phase retrieval, but if J = {2, 3} then x = (0, 1, −1) ⊥ PJxi for +i = 1, 2 and y = (0, 1, 1) ⊥ xi for i = 3, 4 and {PJxi}4 +i=1 fails phase retrieval. +Corollary 3. Assume {xi}2n−2 +i=1 +does weak phase retrieval in Rn and for every J ⊂ [n] +{PJxi}2n−2 +i=1 +does phase retrieval. Then if x, y ∈ Rn and +|⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, . . . , 2n − 2, +then there is a J ⊂ [n] so that +x(j) = +� +aj ̸= 0 for j ∈ J +0 for j ∈ Jc +y(j) = +� +0 for j ∈ J +bj ̸= 0 for j ∈ Jc +Proposition 3. Let {ei}n +i=1 be the unit vector basis of Rn and for I ⊂ [n], let PI be +the projection onto XI = span{ei}i∈I. For every m ≥ 1, there are vectors {xi}m +i=1 +so that for every I ⊂ [1, n], {PIxi}m +i=1 is full spark in XI. +Proof. We do this by induction on m. For m=1, let x1 = (1, 1, 1, . . ., 1). This +satisfies the theorem. So assume the theorem holds for {xi}m +i=1. Choose I ⊂ [1, n] +with |I| = k. Choose J ⊂ I with |J| = k − 1 and let XJ = span{xi}i∈J ∪ {xi}i∈Ic. +Then XJ is a hyperplane in Rn for every J. Since there only exist finitely many +such J′s there is a vector xm+1 /∈ XJ for every J. We will show that {xi}m+1 +i=1 +satisfies the theorem. +Let I ⊂ [1, n] and J ⊂ I with |J| = |I|. If PIxm+1 /∈ XJ, then {PIxi}i∈J is +linearly independent by the induction hypothesis. On the other hand, if m + 1 ∈ J +then xm+1 /∈ XJ. But, if PIxm+1 ∈ span{PIxi}i∈J\m+1, since (I − PI)xm+1 ∈ +span{ei}i∈Ic, it follows that xm+1 ∈ XJ, which is a contradiction. +□ + +10 +P. G. CASAZZA AND F. AKRAMI +Remark 6.2. In the above proposition, none of the xi can have a zero coordinate. +Since if it does, projecting the vectors onto that coordinate produces a zero vector +and so is not full spark. +References +[1] F. Akrami, P. G. Casazza, M. A. Hasankhani Fard, A. Rahimi, A note on norm retrievable +real Hilbert space frames, J. Math. Anal. Appl. 2021. (517)2, (2023) 126620. +[2] P. G. Casazza, F. Akrami, A. Rahimi, fundamental results on weak phase retrieval, Ann. +Funct. Anal, arXiv: 2110.06868, 2021. +[3] S. Bahmanpour, J. Cahill, P.G. Casazza, J. Jasper, and L. M. Woodland, Phase retrieval and +norm retrieval, arXiv:1409.8266, (2014). +[4] R. Balan, P. G. Casazza, D. Edidin, On signal reconstruction without phase, Appl. Comput. +Harmonic Anal. 20, 3, (2006), 345-356. +[5] S. Botelho-Andrade, Peter G. Casazza, D. Cheng, J. Haas, and Tin T. Tran, Phase retrieval +in ℓ2(R), arXiv:1804.01139v1, (2018). +[6] S. Botelho-Andrade, Peter G. Casazza, D. Cheng, J. Haas, and Tin T. Tran, J. C. Tremain, +and Z. Xu, Phase retrieval by hyperplanes, Am. Math. Soc, comtemp. math. 706, (2018), +21-31. +[7] S. Botelho-Andrade, P. G. Casazza, D. Ghoreishi, S. Jose, J. C. Tremain, Weak phase retrieval +and phaseless reconstruction, arXiv:1612.08018, (2016). +[8] S. Botelho-Andrade, P. G. Casazza, H. V. Nguyen, And J. C. Tremain, Phase retrieval versus +phaseless reconstruction, J. Math. Anal. Appl, 436, 1, (2016), 131-137. +[9] J. Cahill, P.G. Casazza, and I. Daubechies, Phase retrieval in infinite dimensional Hilbert +spaces, Transactions of the AMS, Series B, 3, (2016), 63-76. +[10] J. Cahill, P.G. Casazza, J. Peterson and L. Woodland, Phase retrivial by projections, Houston +Journal of Mathematics 42. 2, (2016), 537-558. +[11] P. G. Casazza, D. Ghoreishi, S. Jose, J. C. Tremain, Norm retrieval and phase Retrieval by +projections, Axioms, 6, (2017), 1-15. +[12] P. G. Casazza and G. Kutyniok, Finite Frames, Theory and applications, Birkhauser, (2013). +[13] O. Christensen, An introduction to frames and Riesz bases, Birkhauser, Boston (2003). +[14] R. J. Duffin, A. C. Schaeffer. A class of nonharmonic Fourier series, Trans. Am. Math. Soc, +72, (1952), 341-366. +Department of Mathematics, University of Missouri, Columbia, USA. +Email address: casazzapeter40@gmail.com +Department of Mathematics, University of Maragheh, Maragheh, Iran. +Email address: fateh.akrami@gmail.com + diff --git a/4tAzT4oBgHgl3EQffvxD/content/tmp_files/2301.01456v1.pdf.txt b/4tAzT4oBgHgl3EQffvxD/content/tmp_files/2301.01456v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e403bbb9e40e59a22b17f64d73aff71bdd1c1dc6 --- /dev/null +++ b/4tAzT4oBgHgl3EQffvxD/content/tmp_files/2301.01456v1.pdf.txt @@ -0,0 +1,1639 @@ +Audio-Visual Efficient Conformer for Robust Speech Recognition +Maxime Burchi, Radu Timofte +Computer Vision Lab, CAIDAS, IFI, University of W¨urzburg, Germany +{maxime.burchi,radu.timofte}@uni-wuerzburg.de +Abstract +End-to-end Automatic Speech Recognition (ASR) sys- +tems based on neural networks have seen large improve- +ments in recent years. The availability of large scale hand- +labeled datasets and sufficient computing resources made it +possible to train powerful deep neural networks, reaching +very low Word Error Rate (WER) on academic benchmarks. +However, despite impressive performance on clean audio +samples, a drop of performance is often observed on noisy +speech. In this work, we propose to improve the noise ro- +bustness of the recently proposed Efficient Conformer Con- +nectionist Temporal Classification (CTC)-based architec- +ture by processing both audio and visual modalities. We im- +prove previous lip reading methods using an Efficient Con- +former back-end on top of a ResNet-18 visual front-end and +by adding intermediate CTC losses between blocks. We con- +dition intermediate block features on early predictions us- +ing Inter CTC residual modules to relax the conditional in- +dependence assumption of CTC-based models. We also re- +place the Efficient Conformer grouped attention by a more +efficient and simpler attention mechanism that we call patch +attention. We experiment with publicly available Lip Read- +ing Sentences 2 (LRS2) and Lip Reading Sentences 3 (LRS3) +datasets. Our experiments show that using audio and visual +modalities allows to better recognize speech in the presence +of environmental noise and significantly accelerate training, +reaching lower WER with 4 times less training steps. Our +Audio-Visual Efficient Conformer (AVEC) model achieves +state-of-the-art performance, reaching WER of 2.3% and +1.8% on LRS2 and LRS3 test sets. Code and pretrained +models are available at https://github.com/burchim/AVEC. +1. Introduction +End-to-end Automatic Speech Recognition based on +deep neural networks has become the standard of state-of- +the-art approaches in recent years [25, 47, 18, 16, 17, 31, 7]. +The availability of large scale hand-labeled datasets and suf- +ficient computing resources made it possible to train power- +40 ms rate +Visual Conformer +Stage 2 +20 ms rate +Visual Conformer +Stage 1 +Visual Front-end +Conv3d + ResNet-18 +Audio Front-end +STFT + Conv2d +Audio Conformer +Stage 1 +Audio Conformer +Stage 2 +Audio Conformer +Stage 3 +Audio-Visual +Fusion Module +Audio-Visual +Conformer Stage +Visual +Back-end +Audio +Back-end +80 ms rate +CTC loss +40 ms rate +80 ms rate +80 ms rate +Figure 1: Audio-Visual Efficient Conformer architec- +ture. The model is trained end-to-end using CTC loss and +takes raw audio waveforms and lip movements from the +speaker as inputs. +ful deep neural networks for ASR, reaching very low WER +on academic benchmarks like LibriSpeech [34]. Neural ar- +chitectures like Recurrent Neural Networks (RNN) [15, 19], +Convolution Neural Networks (CNN) [10, 28] and Trans- +formers [12, 23] have successfully been trained from raw +audio waveforms and mel-spectrograms audio features to +transcribe speech to text. +Recently, Gulati et al. [16] +proposed a convolution-augmented transformer architec- +ture (Conformer) to model both local and global dependen- +cies using convolution and attention to reach better speech +recognition performance. Concurrently, Nozaki et al. [33] +arXiv:2301.01456v1 [cs.CV] 4 Jan 2023 + +++improved CTC-based speech recognition by conditioning +intermediate encoder block features on early predictions us- +ing intermediate CTC losses [14]. Burchi et al. [7] also pro- +posed an Efficient Conformer architecture using grouped +attention for speech recognition, lowering the amount of +computation while achieving better performance. Inspired +from computer vision backbones, the Efficient Conformer +encoder is composed of multiple stages where each stage +comprises a number of Conformer blocks to progressively +downsample and project the audio sequence to wider fea- +ture dimensions. +Yet, even if these audio-only approaches are breaking +the state-of-the-art, one major pitfall for using them in the +real-world is the rapid deterioration of performance in the +presence of ambient noise. In parallel to that, Audio Visual +Speech Recognition (AVSR) has recently attracted a lot of +research attention due to its ability to use image process- +ing techniques to aid speech recognition systems. Preced- +ing works have shown that including the visual modality of +lip movements could improve the robustness of ASR sys- +tems with respect to noise while reaching better recognition +performance [41, 42, 36, 1, 45, 29]. Xu et al. [45] pro- +posed a two-stage approach to first separate the target voice +from background noise using the speakers lip movements +and then transcribe the filtered audio signal with the help of +lip movements. Petridis et al. [36] uses a hybrid architec- +ture, training an LSTM-based sequence-to-sequence (S2S) +model with an auxiliary CTC loss using an early fusion +strategy to reach better performance. Ma et al. [29] uses +Conformer back-end networks with ResNet-18 [20] front- +end networks to improve recognition performance. +Other works focus on Visual Speech Recognition (VSR), +only using lip movements to transcribe spoken language +into text [4, 9, 48, 3, 49, 37, 30]. An important line of +research is the use of cross-modal distillation. Afouras et +al. [3] and Zhao et al. [49] proposed to improve the lip read- +ing performance by distilling from an ASR model trained +on a large-scale audio-only corpus while Ma et al. [30] +uses prediction-based auxiliary tasks. Prajwal et al. [37] +also proposed to use sub-words units instead of characters +to transcribe sequences, greatly reducing running time and +memory requirements. Also providing a language prior, re- +ducing the language modelling burden of the model. +In this work we focus on the design of a noise robust +speech recognition architecture processing both audio and +visual modalities. +We use the recently proposed CTC- +based Efficient Conformer architecture [7] and show that +including the visual modality of lip movements can suc- +cessfully improve noise robustness while significantly ac- +celerating training. Our Audio-Visual Efficient Conformer +(AVEC) reaches lower WER using 4 times less training +steps than its audio-only counterpart. +Moreover, we are +the first work to apply intermediate CTC losses between +blocks [27, 33] to improve visual speech recognition perfor- +mance. We show that conditioning intermediate features on +early predictions using Inter CTC residual modules allows +to close the gap in WER between autoregressive and non- +autoregressive AVSR systems based on S2S. This also helps +to counter a common failure case which is that audio-visual +models tend to ignore the visual modality. In this way, we +force pre-fusion layers to learn spatiotemporal features. Fi- +nally, we replace the Efficient Conformer grouped attention +by a more efficient and simpler attention mechanism that +we call patch attention. Patch attention reaches similar per- +formance to grouped attention while having a lower com- +plexity. The contributions of this work are as follows: +• We improve the noise robustness of the recently pro- +posed Efficient Conformer architecture by processing +both audio and visual modalities. +• We condition intermediate Conformer block features +on early predictions using Inter CTC residual modules +to relax the conditional independence assumption of +CTC models. This allows us to close the gap in WER +between autoregressive and non-autoregressive meth- +ods based on S2S. +• We propose to replace the Efficient Conformer +grouped attention by a more efficient and simpler at- +tention mechanism that we call patch attention. Patch +attention reaches similar performance to grouped at- +tention with a lower complexity. +• We experiment on publicly available LRS2 and LRS3 +datasets and reach state-of-the-art results using audio +and visual modalities. +2. Method +In this section, we describe our proposed Audio-Visual +Efficient Conformer network. The model is composed of +4 main components: An audio encoder, a visual encoder, +an audio-visual fusion module and an audio-visual encoder. +The audio and visual encoders are separated into modality +specific front-end networks to transform each input modal- +ity into temporal sequences and Efficient Conformer back- +end networks to model local and global temporal relation- +ships. The full model is trained end-to-end using intermedi- +ate CTC losses between Conformer blocks in addition to the +output CTC layer. The complete architecture of the model +is shown in Figure 1. +2.1. Model Architecture +Audio front-end. +The audio front-end network first +transforms raw audio wave-forms into mel-spectrograms +using a short-time Fourier transform computed over win- +dows of 20ms with a step size of 10ms. 80-dimensional + +mel-scale log filter banks are applied to the resulting fre- +quency features. The mel-spectrograms are processed by +a 2D convolution stem to extract local temporal-frequency +features, resulting in a 20ms frame rate signal. The audio +front-end architecture is shown in Table 1. +Table 1: Audio Front-end architecture, 1.2 Millions param- +eters. Ta denotes the input audio sample length. +Stage +Layers +Output Shape +Fourier +Transf +STFT: 400 window length +160 hop length, 512 ffts +(257, Ta//160 + 1) +Mel +Scale +Mel Scale: 80 mels +(80, Ta//160 + 1) +Stem +Conv2d: 32, 180 filters, 22 stride +(180, 40, Ta//320 + 1) +Proj +Linear, 180 units +(Ta//320 + 1, 180) +Visual front-end. +The visual front-end network [29] +transforms input video frames into temporal sequences. A +3D convolution stem with kernel size 5 × 7 × 7 is first ap- +plied to the video. Each video frame is then processed inde- +pendently using a 2D ResNet-18 [20] with an output spatial +average pooling. Temporal features are then projected to +the back-end network input dimension using a linear layer. +The visual front-end architecture is shown in Table 2. +Table 2: Visual Front-end architecture, 11.3 Millions pa- +rameters. Tv denotes the number of input video frames. +Stage +Layers +Output Shape +Stem +Conv3d: 5 × 72, 64 filters, 1 × 22 stride +MaxPoo3d: 1 × 32, 1 × 22 stride +(64, Tv, 22, 22) +Res 1 +2 × +� +Conv2d: 32, 64 filters +Conv2d: 32, 64 filters +� +(Tv, 64, 22, 22) +Res 2 +2 × +� +Conv2d: 32, 128 filters +Conv2d: 32, 128 filters +� +(Tv, 128, 11, 11) +Res 3 +2 × +� +Conv2d: 32, 256 filters +Conv2d: 32, 256 filters +� +(Tv, 256, 6, 6) +Res 4 +2 × +� +Conv2d: 32, 512 filters +Conv2d: 32, 512 filters +� +(Tv, 512, 3, 3) +Pool +Global Average Pooling +(Tv, 512) +Proj +Linear, 256 units +(Tv, 256) +Back-end networks. The back-end networks use an Ef- +ficient Conformer architecture. +The Efficient Conformer +encoder was proposed in [7], it is composed of several +stages where each stage comprises a number of Conformer +blocks [16] using grouped attention with relative positional +encodings. The temporal sequence is progressively down- +sampled using strided convolutions and projected to wider +feature dimensions, lowering the amount of computation +while achieving better performance. We use 3 stages in the +audio back-end network to downsample the audio signal to +a 80 milliseconds frame rate. Only 2 stages are necessary +to downsample the visual signal to the same frame rate. Ta- +ble 6 shows the hyper-parameter of each back-end network. +Table 3: Back-end networks hyper-parameters. InterCTC +blocks indicates Conformer blocks having a post Inter CTC +residual module. +Network +Visual +Back-end +Audio +Back-end +Audio-Visual +Encoder +Num Params (M) +13.6 +17.9 +15.9 +Num Stages +2 +3 +1 +Blocks per Stage +6, 1 +5, 6, 1 +5 +Total Num Blocks +7 +12 +5 +Stage Feature Dim +256, 360 +180, 256, 360 +360 +Conv Kernel Size +15 +15 +15 +Stage Patch Size +1, 1 +3, 1, 1 +1 +InterCTC Blocks +3, 6 +8, 11 +2 +Audio-visual fusion module. Similar to [36, 29], we +use an early fusion strategy to learn audio-visual features +and reduce model complexity. The acoustic and visual fea- +tures from the back-end networks are concatenated and fed +into a joint feed-forward network. The concatenated fea- +tures of size 2 × dmodel are first expanded using a linear +layer with output size dff = 4 × dmodel, passed through +a Swish activation function [38] and projected back to the +original feature dimension dmodel. +Audio-visual encoder. The audio-visual encoder is a +single stage back-end network composed of 5 Conformer +blocks without downsampling. +The encoder outputs are +then projected to a CTC layer to maximize the sum of prob- +abilities of correct target alignments. +2.2. Patch Attention. +The Efficient Conformer [7] proposed to replace Multi- +Head Self-Attention (MHSA) [44] in earlier encoder lay- +ers with grouped attention. Grouped MHSA reduce atten- +tion complexity by grouping neighbouring temporal ele- +ments along the feature dimension before applying scaled +dot-product attention. Attention having a quadratic com- +putational complexity with respect to the sequence length, +this caused the network to have an asymmetric complexity +with earlier attention layers requiring more flops than latter +layers with shorter sequence length. In this work, we pro- +pose to replace grouped attention with a simpler and more +efficient attention mechanism that we call patch attention +(Figure 2). Similar to the pooling attention proposed by the +Multiscale Vision Transformer (MViT) [13] for video and +image recognition, the patch attention proceed to an average +Table 4: Attention variants complexities including query, +key, value and output linear projections. n and d are the +sequence length and feature dimension respectively. +Attention +Variant +Hyper +Parameter +Full Attention +Complexity +Regular +- +O(n · d2 + n2 · d) +Grouped +Group Size (g) +O(n · d2 + (n/g)2 · d · g) +Patch +Patch Size (k) +O(n/k · d2 + (n/k)2 · d) + +AvgPool +1 +2 +3 +4 +5 +6 +7 +8 +9 +a +a +a +b +b +b +c +c +c +Upsample +a +c +Attention +b +a +c +b +Figure 2: Patch Multi-Head Self-Attention. The input sequence is downsampled using an average pooling before applying +multi-head self-attention. The output sequence is then upsampled via nearest neighbor upsampling, reducing attention com- +plexity from O(n2 · d) to O((n/k)2 · d) where k defines the pooling / upsampling kernel size. Patch attention is equivalent +to regular attention when k = 1. +pooling on the input sequence before projection the query, +key and values. +X = AvgPooling1d(Xin) +(1) +with Q, K, V = XW Q, XW K, XW V +(2) +Where W Q, W K, W V ∈ Rd×d are query, key and value +linear projections parameter matrices. MHSA with relative +sinusoidal positional encoding is then performed at lower +resolution as: +MHSA(X) = Concat (O1, ..., OH) W O +(3) +with Oh = softmax +�QhKT +h + Srel +h +√dh +� +Vh +(4) +Where Srel ∈ Rn×n is a relative position score matrix that +satisfy Srel[i, j] = QiET +j−i. E is the linear projection of +a standard sinusoidal positional encoding matrix with posi- +tions ranging from −(nmax − 1) to (nmax − 1). The atten- +tion output sequence is then projected and up-sampled back +to the initial resolution using nearest neighbor up-sampling. +Xout = UpsampleNearest1d(MHSA(X)) +(5) +In consequence, each temporal element of the same patch +produce the same attention output. Local temporal relation- +ships are only modeled in the convolution modules while +global relationships are modeled by patch attention. We +use 1-dimensional patches in this work but patch attention +Audio back-end Conformer Stage +Module Giga FLOPs +0 +0.1 +0.2 +0.3 +Stage 1 (d=180, n=500) +Stage 2 (d=256, n=250) +Stage 3 (d=360, n=125) +attention +grouped attention (g=3) +patch attention (k=3) +feed-forward +Figure 3: Audio-only back-end modules FLOPs (Billion). +could also be generalized to image and video data using +2D and 3D patches. We leave this to future works. The +computational complexity of each attention variant is shown +in Table 4. Path attention further reduce complexity com- +pared to grouped attention by decreasing the amount of +computation needed by Query, Key, Value and Output fully +connected layers while keeping the feature dimension un- +changed. Similar to previous work [7], we only use patch +attention in the first audio back-end stage to reduce com- +plexity while maintaining model recognition performance. +Figure 3 shows the amount of FLOPs for each attention +module variant with respect to encoded sequence length n +and model feature dimension d. Using patch or grouped at- +tention variants instead of regular MHSA greatly reduce the +amount of FLOPs in the first audio back-end stage. +2.3. Intermediate CTC Predictions. +Inspired by [27] and [33] who proposed to add interme- +diate CTC losses between encoder blocks to improve CTC- +based speech recognition performance, we add Inter CTC +residual modules (Figure 4) in encoder networks. We con- +dition intermediate block features of both audio, visual and +audio-visual encoders on early predictions to relax the con- +ditional independence assumption of CTC models. During +both training and inference, each intermediate prediction is +summed to the input of the next layer to help recognition. +We use the same method proposed in [33] except that we do +not share layer parameters between losses. The lth block +output Xout +l +is passed through a feed-forward network with +residual connection and a softmax activation function: +Zl = Softmax(Linear(Xout +l +)) +(6) +Xin +l+1 = Xout +l ++ Linear(Zl) +(7) +Where Zl ∈ RT ×V is a probability distribution over the +output vocabulary. The intermediate CTC loss is then com- +puted using the target sequence y as: +Linter +l += −log(P(y|Zl)) +(8) +with P(y|Zl) = +� +π∈B−1 +CT C(y) +T +� +t=1 +Zt,πt +(9) + +Conformer Block +Inter CTC +Residual Module +Conformer Block +Linear +Softmax +Linear +CTC loss ++ +Figure 4: Inter CTC residual module. Intermediate pre- +dictions are summed to the input of the next Conformer +block to condition the prediction of the final block on it. +Intermediate CTC losses are added to the output CTC loss +for the computation of the final loss. +Where π ∈ V T are paths of tokens and BCT C is a many-to- +one map that simply removes all blanks and repeated labels +from the paths. The total training objective is defined as +follows: +L = (1 − λ)LCT C + λLinter +(10) +with Linter = 1 +K +� +k∈interblocks +Linter +k +(11) +Where interblocks is the set of blocks having a post Inter +CTC residual module (Figure 4). Similar to [33], we use +Inter CTC residual modules every 3 Conformer blocks with +λ set to 0.5 in every experiments. +3. Experiments +3.1. Datasets +We use 3 publicly available AVSR datasets in this +work. The Lip Reading in the Wild (LRW) [8] dataset is +used for visual pre-training and the Lip Reading Sentences +2 (LRS2) [1] and Lip Reading Sentences 3 (LRS3) [2] +datasets are used for training and evaluation. +LRW dataset. LRW is an audio-visual word recogni- +tion dataset consisting of short video segments containing a +single word out of a vocabulary of 500. The dataset com- +prise 488,766 training samples with at least 800 utterances +per class and a validation and test sets of 25,000 samples +containing 50 utterances per class. +LRS2 & LRS3 datasets. The LRS2 dataset is composed +of 224.1 hours with 144,482 videos clips from the BBC tele- +vision whereas the LRS3 dataset consists of 438.9 hours +with 151,819 video clips extracted from TED and TEDx +talks. Both datasets include corresponding subtitles with +word alignment boundaries and are composed of a pre-train +split, train-val split and test split. LRS2 has 96,318 utter- +ances for pre-training (195 hours), 45,839 for training (28 +hours), 1,082 for validation (0.6 hours), and 1,243 for test- +ing (0.5 hours). Whereas LRS3 has 118,516 utterances in +the pre-training set (408 hours), 31,982 utterances in the +training-validation set (30 hours) and 1,321 utterances in +the test set (0.9 hours). All videos contain a single speaker, +have a 224 × 224 pixels resolution and are sampled at 25 +fps with 16kHz audio. +3.2. Implementation Details +Pre-processing Similar to [29], we remove differences +related to rotation and scale by cropping the lip regions us- +ing bounding boxes of 96 × 96 pixels to facilitate recog- +nition. The RetinaFace [11] face detector and Face Align- +ment Network (FAN) [6] are used to detect 68 facial land- +marks. The cropped images are then converted to gray-scale +and normalised between −1 and 1. Facial landmarks of the +LRW, LRS2 and LRS3 datasets are obtained from previous +work [30] and reused for pre-processing to get a clean com- +parison of the methods. A byte-pair encoding tokenizer is +built from LRS2&3 pre-train and trainval splits transcripts +using sentencepiece [26]. We use a vocabulary size of 256 +including the CTC blank token following preceding works +on CTC-based speech recognition [31, 7]. +Data augmentation Spec-Augment [35] is applied on +the audio mel-spectrograms during training to prevent over- +fitting with two frequency masks with mask size parameter +F = 27 and five time masks with adaptive size pS = 0.05. +Similarly to [30], we mask videos on the time axis using one +mask per second with the maximum mask duration set to 0.4 +seconds. Random cropping with size 88×88 and horizontal +flipping are also performed for each video during training. +We also follow Prajwal et al. [37] using central crop with +horizontal flipping at test time for visual-only experiments. +Training Setup We first pre-train the visual encoder on +the LRW dataset [8] using cross-entropy loss to recognize +words being spoken. The visual encoder is pre-trained for +30 epochs and front-end weights are then used as initializa- +tion for training. Audio and visual encoders are trained on +the LRS2&3 datasets using a Noam schedule [44] with 10k +warmup steps and a peak learning rate of 1e-3. We use the +Adam optimizer [24] with β1 = 0.9, β2 = 0.98. L2 regular- +ization with a 1e-6 weight is also added to all the trainable +weights of the model. We train all models with a global +batch size of 256 on 4 GPUs, using a batch size of 16 per +GPU with 4 accumulated steps. Nvidia A100 40GB GPUs +are used for visual-only and audio-visual experiments while +RTX 2080 Ti are used for audio-only experiments. The +audio-only models are trained for 200 epochs while visual- +only and audio-visual models are trained for 100 and 70 +epochs respectively. Note that we only keep videos shorter +than 400 frames (16 seconds) during training. Finally, we +average models weights over the last 10 epoch checkpoints +using Stochastic Weight Averaging [22] before evaluation. + +Table 5: Comparison of WER (%) on LRS2 / LRS3 test sets with recently published methods using publicly and non-publicly +available datasets for Audio-Only (AO), Visual-Only (VO) and Audio-Visual (AV) models. +Method +Model +Criterion +Training +Datasets +Total +Hours +test WER +AO +VO +AV +(↓) Using Publicly Available Datasets (↓) +Petridis et al. [36] +CTC+S2S +LRW, LRS2 +381 +8.3 / - +63.5 / - +7.0 / - +Zhang et al. [48] +S2S +LRW, LRS2&3 +788 / 790 +- +51.7 / 60.1 +- +Afouras et al. [3] +CTC +VoxCeleb2clean, LRS2&3 +1,032 / 808 +- +51.3 / 59.8 +- +Xu et al. [45] +S2S +LRW, LRS3 +595 +- / 7.2 +- / 57.8 +- / 6.8 +Yu et al.[46] +LF-MMI +LRS2 +224 +6.7 / - +48.9 / - +5.9 / - +Ma et al. [29] +CTC+S2S +LRW, LRS2&3 +381 / 595 +3.9 / 2.3 +37.9 / 43.3 +3.7 / 2.3 +Prajwal et al. [37] +S2S +LRS2&3 +698 +- +28.9 / 40.6 +- +Ma et al. [30] +CTC+S2S +LRW, LRS2&3 +818 +- +27.3 / 34.7 +- +Ours +CTC +LRW, LRS2&3 +818 +2.8 / 2.1 +32.6 / 39.2 +2.5 / 1.9 ++ Neural LM +CTC +LRW, LRS2&3 +818 +2.4 / 2.0 +29.8 / 37.5 +2.3 / 1.8 +(↓) Using Non-Publicly Available Datasets (↓) +Afouras et al. [1] +S2S +MVLRS, LRS2&3 +1,395 +9.7 / 8.3 +48.3 / 58.9 +8.5 / 7.2 +Zhao et al. [49] +S2S +MVLRS, LRS2 +954 +- +65.3 / - +- +Shillingford et al. [40] +CTC +LRVSR +3,886 +- +- / 55.1 +- +Makino et al. [32] +Transducer +YouTube-31k +31,000 +- / 4.8 +- / 33.6 +- / 4.5 +Serdyuk et al. [39] +Transducer +YouTube-90k +91,000 +- +- / 25.9 +- / 2.3 +Prajwal et al. [37] +S2S +MVLRS, TEDxext, LRS2&3 +2,676 +- +22.6 / 30.7 +- +Ma et al. [30] +CTC+S2S +LRW, AVSpeech, LRS2&3 +1,459 +- +25.5 / 31.5 +- +Language Models. Similarly to [28], we experiment +with a N-gram [21] statistical language model (LM) and a +Transformer neural language model. A 6-gram LM is used +to generate a list of hypotheses using beam search and an +external Transformer LM is used to rescore the final list. +The 6-gram LM is trained on the LRS2&3 pre-train and +train-val transcriptions. Concerning the neural LM, we pre- +train a 12 layer GPT-3 Small [5] on the LibriSpeech LM +corpus for 0.5M steps using a batch size of 0.1M tokens +and finetune it for 10 epochs on the LRS2&3 transcriptions. +3.3. Results +Table 5 compares WERs of our Audio-Visual Effi- +cient Conformer with state-of-the-art methods on the LRS2 +and LRS3 test sets. +Our Audio-Visual Efficient Con- +former achieves state-of-the-art performances with WER of +2.3%/1.8%. On the visual-only track, our CTC model com- +petes with most recent autoregressive methods using S2S +criterion. We were able to recover similar results but still +lack behind Ma et al. [30] which uses auxiliary losses with +pre-trained audio-only and visual-only networks. We found +our audio-visual network to converge faster than audio-only +experiments, reaching better performance using 4 times less +training steps. The intermediate CTC losses of the visual +encoder could reach lower levels than in visual-only experi- +ments showing that optimizing audio-visual layers can help +pre-fusion layers to learn better representations. +3.4. Ablation Studies +We propose a detailed ablation study to better understand +the improvements in terms of complexity and WER brought +by each architectural modification. We report the number +of operations measured in FLOPs (number of multiply-and- +add operations) for the network to process a ten second au- +dio/video clip. Inverse Real Time Factor (Inv RTF) is also +measured on the LRS3 test set by decoding with a batch +size 1 on a single Intel Core i7-12700 CPU thread. All abla- +tions were performed by training audio-only models for 200 +epochs and visual-only / audio-visual models for 50 epochs. +Efficient Conformer Visual Back-end. We improve the +recently proposed visual Conformer encoder [29] using an +Efficient Conformer back-end network. The use of byte pair +encodings for tokenization instead of characters allows us to +further downsample temporal sequences without impacting +the computation of CTC loss. Table 6 shows that using an +Efficient Conformer back-end network for our visual-only +model leads to better performances while reducing model +complexity and training time. The number of model param- +eters is also slightly decreased. +Table 6: Ablation study on visual back-end network. +Visual +Back-end +#Params +(Million) +LRS2 +test +LRS3 +test +#FLOPs +(Billion) +Inv +RTF +Conformer +43.0 +39.53 +47.14 +87.94 +5.17 +Eff Conf +40.4 +37.39 +44.96 +84.52 +5.26 + +Reference +the authors looked at papers written over a 10 year period and hundreds had to be thrown out +Outputs +Block 3: the otho looing pa people we over s any your per and conndries that aboutent threghow +Block 6: the autthherss looking paperss we overai year paiod and hundreds that about thrououtow +Block 9: the authors looked at papers witen over ainght year period and hundreds that to been throw out +Block 12: the authors looked at papers written over 10 year period and hundreds had to be thrown out +Figure 5: Output example of our Visual-only model using greedy search decoding on the LRS3 test set with intermediate +CTC prediction every 3 blocks. The sentence is almost correctly transcribed except for the missing ’a’ before ’10 year’. +Inter CTC residual modules. Similar to [33], we exper- +iment adding Inter CTC residual modules between blocks +to relax the conditional independence assumption of CTC. +Table 7 shows that using intermediate CTC losses every 3 +Conformer blocks greatly helps to reduce WER, except for +the audio-only setting where this does not improve perfor- +mance. Figure 5 gives an example of intermediate block +predictions decoded using greedy search without an exter- +nal language model on the test set of LRS3. We can see +that the output is being refined in the encoder layers by con- +ditioning on the intermediate predictions of previous lay- +ers. Since our model refines the output over the frame-level +predictions, it can correct insertion and deletion errors in +addition to substitution errors. We further study the im- +pact of Inter CTC on multi-modal learning by measuring +the performance of our audio-visual model when one of +the two modalities is masked. As pointed out by preced- +ing works [8, 1, 32], networks with multi-modal inputs can +often be dominated by one of the modes. In our case speech +recognition is a significantly easier problem than lip reading +which can cause the model to ignore visual information. Ta- +ble 8 shows that Inter CTC can help to counter this problem +by forcing pre-fusion layers to transcribe the input signal. +Table 7: Ablation study on Inter CTC residual modules. +Model +Back-end +#Params +(Million) +LRS2 +test +LRS3 +test +#FLOPs +(Billion) +Inv +RTF +Audio-only (↓) +Eff Conf +31.5 +2.83 +2.13 +7.54 +51.98 ++ Inter CTC +32.1 +2.84 +2.11 +7.67 +50.30 +Visual-only (↓) +Eff Conf +40.4 +37.39 +44.96 +84.52 +5.26 ++ Inter CTC +40.9 +33.82 +40.63 +84.60 +5.26 +Audio-visual (↓) +Eff Conf +60.9 +2.87 +2.54 +90.53 +4.84 ++ Inter CTC +61.7 +2.58 +1.99 +90.66 +4.82 +Table 8: Impact of Inter CTC on audio-visual model WER +(%) for LRS2 / LRS3 test sets in a masked modality setting. +Inter CTC +Audio-Visual Eval Mode +masked video +masked audio +no mask +No +4.48 / 3.22 +52.77 / 59.10 +2.87 / 2.54 +Yes +3.39 / 2.38 +37.62 / 46.55 +2.58 / 1.99 +Patch multi-head self-attention. +We experiment re- +placing grouped attention by patch attention in the first +audio encoder stage. Our objective being to increase the +model efficiency and simplicity without harming perfor- +mance. Grouped attention was proposed in [7] to reduce +attention complexity for long sequences in the first encoder +stage. Table 9 shows the impact of each attention variant +on our audio-only model performance and complexity. We +start with an Efficient Conformer (M) [7] and replace the +attention mechanism. We find that grouped attention can be +replaced by patch attention without a loss of performance +using a patch size of 3 in the first back-end stage. +Table 9: Ablation study on audio back-end attention. +Attention +Type +Group / +Patch Size +LRS2 +test +LRS3 +test +#FLOPs +(Billion) +Inv +RTF +Regular +- +2.85 +2.12 +8.66 +49.86 +Grouped +3, 1, 1 +2.82 +2.13 +8.06 +50.27 +Patch +3, 1, 1 +2.83 +2.13 +7.54 +51.98 +3.5. Noise Robustness +We measure model noise robustness using various types +of noise and compare our Audio-Visual Efficient Conformer +with recently published methods. Figure 6 shows the WER +evolution of audio-only (AO), visual-only (VO) and audio- +visual (AV) models with respect to multiple Signal to Noise +Ratio (SNR) using white noise and babble noise from the +NoiseX corpus [43]. We find that processing both audio and +visual modalities can help to significantly improve speech +recognition robustness with respect to babble noise. More- +over, we also experiment adding babble noise during train- +ing as done in previous works [36, 29] and find that it can +further improve noise robustness at test time. +Robustness to various types of noise. We gather var- +ious types of recorded audio noise including sounds and +music. In Table 10, we observe that the Audio-Visual Ef- +ficient Conformer consistently achieves better performance +than its audio-only counterpart in the presence of various +noise types. This confirm our hypothesis that the audio- +visual model is able to use the visual modality to aid speech +recognition when audio noise is present in the input. + +SNR (dB) +Word Error Rate (%) +0 +10 +20 +30 +40 +50 +-5 +0 +5 +10 +15 +20 +VO LRS2 +AO LRS2 +AV LRS2 +AV* LRS2 +VO LRS3 +AO LRS3 +AV LRS3 +AV* LRS3 +(a) Babble noise +SNR (dB) +Word Error Rate (%) +0 +10 +20 +30 +40 +50 +-5 +0 +5 +10 +15 +20 +VO LRS2 +AO LRS2 +AV LRS2 +AV* LRS2 +VO LRS3 +AO LRS3 +AV LRS3 +AV* LRS3 +(b) White noise +Figure 6: LRS2 and LRS3 test WER (%) as a function +of SNR (dB). * indicates experiments being trained with +babble noise. We measure noise robustness by evaluating +our models in the presence of babble and white noise. +Table 10: LRS3 test WER (%) as a function of SNR (dB). +Noise +Mode +SNR (dB) +-5 +0 +5 +10 +15 +20 +babble +AO +75.9 +32.4 +9.3 +4.1 +2.7 +2.3 +AV +33.5 +14.8 +5.4 +3.0 +2.3 +2.0 +AV* +11.2 +4.9 +3.1 +2.5 +2.2 +2.0 +white +AO +77.6 +34.0 +15.5 +7.3 +4.1 +2.8 +AV +28.9 +14.7 +5.5 +3.0 +2.3 +2.0 +AV* +17.4 +8.9 +3.6 +2.8 +2.3 +2.0 +birds +AO +51.8 +23.9 +10.9 +5.9 +3.7 +2.8 +AV +21.6 +11.5 +6.2 +4.1 +2.9 +2.4 +AV* +15.9 +8.3 +4.9 +3.4 +2.7 +2.4 +chainsaw +AO +82.9 +41.2 +14.8 +5.5 +3.7 +2.7 +AV +37.8 +17.3 +7.6 +3.9 +2.6 +2.3 +AV* +25.8 +10.8 +5.0 +3.2 +2.4 +2.3 +jazz +AO +25.3 +9.7 +4.1 +3.1 +2.6 +2.3 +AV +13.9 +6.0 +3.2 +2.4 +2.3 +2.0 +AV* +10.6 +4.2 +2.8 +2.4 +2.2 +2.0 +street +raining +AO +58.4 +23.8 +8.9 +4.6 +3.0 +2.5 +AV +27.12 +10.8 +5.7 +3.1 +2.7 +2.3 +AV* +15.9 +6.9 +3.8 +2.7 +2.3 +2.2 +washing +dishes +AO +47.8 +24.5 +11.5 +6.0 +3.7 +2.8 +AV +21.3 +11.5 +6.1 +3.6 +2.8 +2.3 +AV* +14.2 +7.3 +4.3 +2.2 +2.6 +2.3 +train +AO +51.3 +18.6 +7.0 +4.0 +2.9 +2.5 +AV +23.1 +10.1 +4.7 +3.0 +2.4 +2.2 +AV* +14.5 +6.2 +3.5 +2.6 +2.3 +2.2 +Comparison with other methods. +We compare our +method with results provided by Ma et al. [29] and +Petridis et al. [36] on the LRS2 test set. Table 11 shows that +our audio-visual model achieves lower WER in the pres- +ence of babble noise, reaching WER of 9.7% at -5 dB SNR +against 16.3% for Ma et al. [29]. +Table 11: Comparison with Ma et al. [29]. LRS2 test WER +(%) as a function of SNR (dB) using babble noise. +Method +Mode +SNR (dB) +-5 +0 +5 +10 +15 +20 +Ma et al. [29] +VO +37.9 +37.9 +37.9 +37.9 +37.9 +37.9 +AO* +28.8 +9.8 +7 +5.2 +4.5 +4.2 +AV* +16.3 +7.5 +6.1 +4.7 +4.4 +4.2 +Ours +VO +32.6 +32.6 +32.6 +32.6 +32.6 +32.6 +AO +70.5 +27 +8.6 +4.7 +3.4 +3.1 +AV +25 +11.2 +5.1 +3.2 +2.8 +2.6 +AV* +9.7 +5 +3.4 +2.9 +2.8 +2.6 +Table 12: Comparison with Petridis et al. [36]. LRS2 test +WER (%) as a function of SNR (dB) using white noise. +Method +Mode +SNR (dB) +-5 +0 +5 +10 +15 +20 +Petridis et al. [36] +VO +63.5 +63.5 +63.5 +63.5 +63.5 +63.5 +AO* +85.0 +45.4 +19.6 +11.7 +9.4 +8.4 +AV* +55.0 +26.1 +13.2 +9.4 +8.0 +7.3 +Ours +VO +32.6 +32.6 +32.6 +32.6 +32.6 +32.6 +AO +73.1 +32.3 +14.3 +7.2 +4.4 +3.5 +AV +22.5 +11.5 +6.2 +4.1 +3.2 +2.9 +AV* +14.4 +8.0 +5.1 +3.9 +3.1 +2.9 +4. Conclusion +In this paper, we proposed to improve the noise robust- +ness of the recently proposed Efficient Conformer CTC- +based architecture by processing both audio and visual +modalities. We showed that incorporating multi-scale CTC +losses between blocks could help to improve recognition +performance, reaching comparable results to most recent +autoregressive lip reading methods. We also proposed patch +attention, a simpler and more efficient attention mechanism +to replace grouped attention in the first audio encoder stage. +Our Audio-Visual Efficient Conformer achieves state-of- +the-art performance of 2.3% and 1.8% on the LRS2 and +LRS3 test sets. +In the future, we would like to explore +other techniques to further improve the noise robustness +of our model and close the gap between recent lip reading +methods. This includes adding various audio noises during +training and using cross-modal distillation with pre-trained +models. We also wish to reduce the visual front-end net- +work complexity without arming recognition performance +and experiment with the RNN-Transducer learning objec- +tive for streaming applications. +Acknowledgments +This work was partly supported by The Alexander von +Humboldt Foundation (AvH). + +References +[1] Triantafyllos Afouras, Joon Son Chung, Andrew Senior, +Oriol Vinyals, and Andrew Zisserman. Deep audio-visual +speech recognition. IEEE transactions on pattern analysis +and machine intelligence, 2018. +[2] Triantafyllos Afouras, Joon Son Chung, and Andrew Zisser- +man. Lrs3-ted: a large-scale dataset for visual speech recog- +nition. arXiv preprint arXiv:1809.00496, 2018. +[3] Triantafyllos Afouras, Joon Son Chung, and Andrew Zisser- +man. Asr is all you need: Cross-modal distillation for lip +reading. In ICASSP 2020-2020 IEEE International Confer- +ence on Acoustics, Speech and Signal Processing (ICASSP), +pages 2143–2147. IEEE, 2020. +[4] Yannis M Assael, Brendan Shillingford, Shimon Whiteson, +and Nando De Freitas. Lipnet: End-to-end sentence-level +lipreading. arXiv preprint arXiv:1611.01599, 2016. +[5] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Sub- +biah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakan- +tan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Lan- +guage models are few-shot learners. Advances in neural in- +formation processing systems, 33:1877–1901, 2020. +[6] Adrian Bulat and Georgios Tzimiropoulos. How far are we +from solving the 2d & 3d face alignment problem?(and a +dataset of 230,000 3d facial landmarks). +In Proceedings +of the IEEE International Conference on Computer Vision, +pages 1021–1030, 2017. +[7] Maxime Burchi and Valentin Vielzeuf. Efficient conformer: +Progressive downsampling and grouped attention for auto- +matic speech recognition. In 2021 IEEE Automatic Speech +Recognition and Understanding Workshop (ASRU), pages 8– +15. IEEE, 2021. +[8] Joon Son Chung and Andrew Zisserman. Lip reading in the +wild. In Asian conference on computer vision, pages 87–103. +Springer, 2016. +[9] Joon Son Chung and AP Zisserman. Lip reading in profile. +2017. +[10] Ronan Collobert, Christian Puhrsch, and Gabriel Synnaeve. +Wav2letter: an end-to-end convnet-based speech recognition +system. arXiv preprint arXiv:1609.03193, 2016. +[11] Jiankang Deng, Jia Guo, Evangelos Ververas, Irene Kot- +sia, and Stefanos Zafeiriou. Retinaface: Single-shot multi- +level face localisation in the wild. +In Proceedings of +the IEEE/CVF conference on computer vision and pattern +recognition, pages 5203–5212, 2020. +[12] Linhao Dong, Shuang Xu, and Bo Xu. Speech-transformer: +a no-recurrence sequence-to-sequence model for speech +recognition. +In 2018 IEEE International Conference on +Acoustics, Speech and Signal Processing (ICASSP), pages +5884–5888. IEEE, 2018. +[13] Haoqi Fan, Bo Xiong, Karttikeya Mangalam, Yanghao Li, +Zhicheng Yan, Jitendra Malik, and Christoph Feichten- +hofer. +Multiscale vision transformers. +In Proceedings of +the IEEE/CVF International Conference on Computer Vi- +sion, pages 6824–6835, 2021. +[14] Alex Graves, Santiago Fern´andez, Faustino Gomez, and +J¨urgen Schmidhuber. Connectionist temporal classification: +labelling unsegmented sequence data with recurrent neural +networks. In ICML, pages 369–376, 2006. +[15] Alex Graves, Abdel-rahman Mohamed, and Geoffrey Hin- +ton. Speech recognition with deep recurrent neural networks. +In 2013 IEEE international conference on acoustics, speech +and signal processing, pages 6645–6649. Ieee, 2013. +[16] Anmol Gulati, James Qin, Chung-Cheng Chiu, Niki Par- +mar, Yu Zhang, Jiahui Yu, Wei Han, Shibo Wang, Zheng- +dong Zhang, Yonghui Wu, et al. Conformer: Convolution- +augmented transformer for speech recognition. +arXiv +preprint arXiv:2005.08100, 2020. +[17] Pengcheng Guo, Florian Boyer, Xuankai Chang, Tomoki +Hayashi, Yosuke Higuchi, Hirofumi Inaguma, Naoyuki +Kamo, Chenda Li, Daniel Garcia-Romero, Jiatong Shi, et al. +Recent developments on espnet toolkit boosted by con- +former. In ICASSP 2021-2021 IEEE International Confer- +ence on Acoustics, Speech and Signal Processing (ICASSP), +pages 5874–5878. IEEE, 2021. +[18] Wei Han, Zhengdong Zhang, Yu Zhang, Jiahui Yu, Chung- +Cheng Chiu, James Qin, Anmol Gulati, Ruoming Pang, and +Yonghui Wu. Contextnet: Improving convolutional neural +networks for automatic speech recognition with global con- +text. arXiv preprint arXiv:2005.03191, 2020. +[19] Awni Hannun, Carl Case, Jared Casper, Bryan Catanzaro, +Greg Diamos, Erich Elsen, Ryan Prenger, Sanjeev Satheesh, +Shubho Sengupta, Adam Coates, et al. +Deep speech: +Scaling up end-to-end speech recognition. +arXiv preprint +arXiv:1412.5567, 2014. +[20] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. +Deep residual learning for image recognition. In Proceed- +ings of the IEEE conference on computer vision and pattern +recognition, pages 770–778, 2016. +[21] Kenneth Heafield. +Kenlm: Faster and smaller language +model queries. In Proceedings of the sixth workshop on sta- +tistical machine translation, pages 187–197, 2011. +[22] Pavel Izmailov, Dmitrii Podoprikhin, Timur Garipov, Dmitry +Vetrov, and Andrew Gordon Wilson. +Averaging weights +leads to wider optima and better generalization. +arXiv +preprint arXiv:1803.05407, 2018. +[23] Shigeki Karita, Nanxin Chen, Tomoki Hayashi, Takaaki +Hori, Hirofumi Inaguma, Ziyan Jiang, Masao Someki, +Nelson Enrique Yalta Soplin, Ryuichi Yamamoto, Xiaofei +Wang, et al. A comparative study on transformer vs rnn in +speech applications. In 2019 IEEE Automatic Speech Recog- +nition and Understanding Workshop (ASRU), pages 449– +456. IEEE, 2019. +[24] Diederik P Kingma and Jimmy Ba. Adam: A method for +stochastic optimization. +arXiv preprint arXiv:1412.6980, +2014. +[25] Samuel Kriman, Stanislav Beliaev, Boris Ginsburg, Joce- +lyn Huang, Oleksii Kuchaiev, Vitaly Lavrukhin, Ryan Leary, +Jason Li, and Yang Zhang. +Quartznet: Deep automatic +speech recognition with 1d time-channel separable convolu- +tions. In ICASSP 2020-2020 IEEE International Conference +on Acoustics, Speech and Signal Processing (ICASSP), pages +6124–6128. IEEE, 2020. +[26] Taku Kudo and John Richardson. Sentencepiece: A sim- +ple and language independent subword tokenizer and detok- + +enizer for neural text processing. In EMNLP, pages 66–71, +2018. +[27] Jaesong Lee and Shinji Watanabe. Intermediate loss regular- +ization for ctc-based speech recognition. In ICASSP 2021- +2021 IEEE International Conference on Acoustics, Speech +and Signal Processing (ICASSP), pages 6224–6228. IEEE, +2021. +[28] Jason Li, Vitaly Lavrukhin, Boris Ginsburg, Ryan Leary, +Oleksii Kuchaiev, Jonathan M Cohen, Huyen Nguyen, and +Ravi Teja Gadde. Jasper: An end-to-end convolutional neu- +ral acoustic model. arXiv preprint arXiv:1904.03288, 2019. +[29] Pingchuan Ma, Stavros Petridis, and Maja Pantic. +End- +to-end audio-visual speech recognition with conformers. +In ICASSP 2021-2021 IEEE International Conference on +Acoustics, Speech and Signal Processing (ICASSP), pages +7613–7617. IEEE, 2021. +[30] Pingchuan Ma, Stavros Petridis, and Maja Pantic. +Visual +speech recognition for multiple languages in the wild. arXiv +preprint arXiv:2202.13084, 2022. +[31] Somshubra Majumdar, Jagadeesh Balam, Oleksii Hrinchuk, +Vitaly Lavrukhin, Vahid Noroozi, and Boris Ginsburg. Cit- +rinet: Closing the gap between non-autoregressive and au- +toregressive end-to-end models for automatic speech recog- +nition. arXiv preprint arXiv:2104.01721, 2021. +[32] Takaki Makino, +Hank Liao, +Yannis Assael, +Brendan +Shillingford, Basilio Garcia, Otavio Braga, and Olivier Sio- +han. Recurrent neural network transducer for audio-visual +speech recognition. In 2019 IEEE automatic speech recog- +nition and understanding workshop (ASRU), pages 905–912. +IEEE, 2019. +[33] Jumon Nozaki and Tatsuya Komatsu. +Relaxing the con- +ditional independence assumption of ctc-based asr by con- +ditioning on intermediate predictions. +arXiv preprint +arXiv:2104.02724, 2021. +[34] Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev +Khudanpur. Librispeech: an asr corpus based on public do- +main audio books. In 2015 IEEE international conference +on acoustics, speech and signal processing (ICASSP), pages +5206–5210. IEEE, 2015. +[35] Daniel S Park, Yu Zhang, Chung-Cheng Chiu, Youzheng +Chen, Bo Li, William Chan, Quoc V Le, and Yonghui Wu. +Specaugment on large scale datasets. +In ICASSP, pages +6879–6883, 2020. +[36] Stavros Petridis, Themos Stafylakis, Pingchuan Ma, Geor- +gios Tzimiropoulos, and Maja Pantic. Audio-visual speech +recognition with a hybrid ctc/attention architecture. In 2018 +IEEE Spoken Language Technology Workshop (SLT), pages +513–520. IEEE, 2018. +[37] KR Prajwal, Triantafyllos Afouras, and Andrew Zisserman. +Sub-word level lip reading with visual attention. In Proceed- +ings of the IEEE/CVF Conference on Computer Vision and +Pattern Recognition, pages 5162–5172, 2022. +[38] Prajit Ramachandran, +Barret Zoph, +and Quoc V Le. +Searching +for +activation +functions. +arXiv +preprint +arXiv:1710.05941, 2017. +[39] Dmitriy Serdyuk, Otavio Braga, and Olivier Siohan. Audio- +visual speech recognition is worth 32x32x8 voxels. +In +2021 IEEE Automatic Speech Recognition and Understand- +ing Workshop (ASRU), pages 796–802. IEEE, 2021. +[40] Brendan Shillingford, Yannis Assael, Matthew W Hoff- +man, Thomas Paine, C´ıan Hughes, Utsav Prabhu, Hank +Liao, Hasim Sak, Kanishka Rao, Lorrayne Bennett, et al. +Large-scale visual speech recognition. +arXiv preprint +arXiv:1807.05162, 2018. +[41] Joon Son Chung, Andrew Senior, Oriol Vinyals, and Andrew +Zisserman. Lip reading sentences in the wild. In Proceed- +ings of the IEEE conference on computer vision and pattern +recognition, pages 6447–6456, 2017. +[42] George Sterpu, Christian Saam, and Naomi Harte. Attention- +based audio-visual fusion for robust automatic speech recog- +nition. In Proceedings of the 20th ACM International Con- +ference on Multimodal Interaction, pages 111–115, 2018. +[43] Andrew Varga and Herman JM Steeneken. Assessment for +automatic speech recognition: Ii. noisex-92: A database and +an experiment to study the effect of additive noise on speech +recognition systems. +Speech communication, 12(3):247– +251, 1993. +[44] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- +reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia +Polosukhin. Attention is all you need. Advances in neural +information processing systems, 30, 2017. +[45] Bo Xu, Cheng Lu, Yandong Guo, and Jacob Wang. Discrim- +inative multi-modality speech recognition. In Proceedings of +the IEEE/CVF Conference on Computer Vision and Pattern +Recognition, pages 14433–14442, 2020. +[46] Jianwei Yu, Shi-Xiong Zhang, Jian Wu, Shahram Ghorbani, +Bo Wu, Shiyin Kang, Shansong Liu, Xunying Liu, Helen +Meng, and Dong Yu. Audio-visual recognition of overlapped +speech for the lrs2 dataset. In ICASSP 2020-2020 IEEE In- +ternational Conference on Acoustics, Speech and Signal Pro- +cessing (ICASSP), pages 6984–6988. IEEE, 2020. +[47] Qian Zhang, Han Lu, Hasim Sak, Anshuman Tripathi, Erik +McDermott, Stephen Koo, and Shankar Kumar. Transformer +transducer: A streamable speech recognition model with +transformer encoders and rnn-t loss. In ICASSP 2020-2020 +IEEE International Conference on Acoustics, Speech and +Signal Processing (ICASSP), pages 7829–7833. IEEE, 2020. +[48] Xingxuan Zhang, Feng Cheng, and Shilin Wang. +Spatio- +temporal fusion based convolutional sequence learning for +lip reading. In Proceedings of the IEEE/CVF International +Conference on Computer Vision, pages 713–722, 2019. +[49] Ya Zhao, Rui Xu, Xinchao Wang, Peng Hou, Haihong Tang, +and Mingli Song. Hearing lips: Improving lip reading by dis- +tilling speech recognizers. In Proceedings of the AAAI Con- +ference on Artificial Intelligence, volume 34, pages 6917– +6924, 2020. + diff --git a/4tAzT4oBgHgl3EQffvxD/content/tmp_files/load_file.txt b/4tAzT4oBgHgl3EQffvxD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..69f9817dac2365876501dd6b3847a184037f5b36 --- /dev/null +++ b/4tAzT4oBgHgl3EQffvxD/content/tmp_files/load_file.txt @@ -0,0 +1,906 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf,len=905 +page_content='Audio-Visual Efficient Conformer for Robust Speech Recognition Maxime Burchi, Radu Timofte Computer Vision Lab, CAIDAS, IFI, University of W¨urzburg, Germany {maxime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='burchi,radu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='timofte}@uni-wuerzburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='de Abstract End-to-end Automatic Speech Recognition (ASR) sys- tems based on neural networks have seen large improve- ments in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The availability of large scale hand- labeled datasets and sufficient computing resources made it possible to train powerful deep neural networks, reaching very low Word Error Rate (WER) on academic benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' However, despite impressive performance on clean audio samples, a drop of performance is often observed on noisy speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In this work, we propose to improve the noise ro- bustness of the recently proposed Efficient Conformer Con- nectionist Temporal Classification (CTC)-based architec- ture by processing both audio and visual modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We im- prove previous lip reading methods using an Efficient Con- former back-end on top of a ResNet-18 visual front-end and by adding intermediate CTC losses between blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We con- dition intermediate block features on early predictions us- ing Inter CTC residual modules to relax the conditional in- dependence assumption of CTC-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We also re- place the Efficient Conformer grouped attention by a more efficient and simpler attention mechanism that we call patch attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We experiment with publicly available Lip Read- ing Sentences 2 (LRS2) and Lip Reading Sentences 3 (LRS3) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Our experiments show that using audio and visual modalities allows to better recognize speech in the presence of environmental noise and significantly accelerate training, reaching lower WER with 4 times less training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Our Audio-Visual Efficient Conformer (AVEC) model achieves state-of-the-art performance, reaching WER of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8% on LRS2 and LRS3 test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Code and pretrained models are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='com/burchim/AVEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Introduction End-to-end Automatic Speech Recognition based on deep neural networks has become the standard of state-of- the-art approaches in recent years [25, 47, 18, 16, 17, 31, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='The availability of large scale hand-labeled datasets and suf- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='ficient computing resources made it possible to train power- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='40 ms rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Visual Conformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Stage 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='20 ms rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Visual Conformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Stage 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Visual Front-end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Conv3d + ResNet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Audio Front-end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='STFT + Conv2d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Audio Conformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Stage 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Audio Conformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Stage 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Audio Conformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Stage 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Audio-Visual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Fusion Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Audio-Visual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Conformer Stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Visual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Back-end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Audio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Back-end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='80 ms rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='CTC loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='40 ms rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='80 ms rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='80 ms rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Figure 1: Audio-Visual Efficient Conformer architec- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The model is trained end-to-end using CTC loss and takes raw audio waveforms and lip movements from the speaker as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' ful deep neural networks for ASR, reaching very low WER on academic benchmarks like LibriSpeech [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Neural ar- chitectures like Recurrent Neural Networks (RNN) [15, 19], Convolution Neural Networks (CNN) [10, 28] and Trans- formers [12, 23] have successfully been trained from raw audio waveforms and mel-spectrograms audio features to transcribe speech to text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Recently, Gulati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [16] proposed a convolution-augmented transformer architec- ture (Conformer) to model both local and global dependen- cies using convolution and attention to reach better speech recognition performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Concurrently, Nozaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [33] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='01456v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='CV] 4 Jan 2023 ++improved CTC-based speech recognition by conditioning intermediate encoder block features on early predictions us- ing intermediate CTC losses [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Burchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [7] also pro- posed an Efficient Conformer architecture using grouped attention for speech recognition, lowering the amount of computation while achieving better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Inspired from computer vision backbones, the Efficient Conformer encoder is composed of multiple stages where each stage comprises a number of Conformer blocks to progressively downsample and project the audio sequence to wider fea- ture dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Yet, even if these audio-only approaches are breaking the state-of-the-art, one major pitfall for using them in the real-world is the rapid deterioration of performance in the presence of ambient noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In parallel to that, Audio Visual Speech Recognition (AVSR) has recently attracted a lot of research attention due to its ability to use image process- ing techniques to aid speech recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Preced- ing works have shown that including the visual modality of lip movements could improve the robustness of ASR sys- tems with respect to noise while reaching better recognition performance [41, 42, 36, 1, 45, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [45] pro- posed a two-stage approach to first separate the target voice from background noise using the speakers lip movements and then transcribe the filtered audio signal with the help of lip movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Petridis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [36] uses a hybrid architec- ture, training an LSTM-based sequence-to-sequence (S2S) model with an auxiliary CTC loss using an early fusion strategy to reach better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [29] uses Conformer back-end networks with ResNet-18 [20] front- end networks to improve recognition performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Other works focus on Visual Speech Recognition (VSR), only using lip movements to transcribe spoken language into text [4, 9, 48, 3, 49, 37, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' An important line of research is the use of cross-modal distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Afouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [3] and Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [49] proposed to improve the lip read- ing performance by distilling from an ASR model trained on a large-scale audio-only corpus while Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [30] uses prediction-based auxiliary tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Prajwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [37] also proposed to use sub-words units instead of characters to transcribe sequences, greatly reducing running time and memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Also providing a language prior, re- ducing the language modelling burden of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In this work we focus on the design of a noise robust speech recognition architecture processing both audio and visual modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We use the recently proposed CTC- based Efficient Conformer architecture [7] and show that including the visual modality of lip movements can suc- cessfully improve noise robustness while significantly ac- celerating training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Our Audio-Visual Efficient Conformer (AVEC) reaches lower WER using 4 times less training steps than its audio-only counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Moreover, we are the first work to apply intermediate CTC losses between blocks [27, 33] to improve visual speech recognition perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We show that conditioning intermediate features on early predictions using Inter CTC residual modules allows to close the gap in WER between autoregressive and non- autoregressive AVSR systems based on S2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' This also helps to counter a common failure case which is that audio-visual models tend to ignore the visual modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In this way, we force pre-fusion layers to learn spatiotemporal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Fi- nally, we replace the Efficient Conformer grouped attention by a more efficient and simpler attention mechanism that we call patch attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Patch attention reaches similar per- formance to grouped attention while having a lower com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The contributions of this work are as follows: We improve the noise robustness of the recently pro- posed Efficient Conformer architecture by processing both audio and visual modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We condition intermediate Conformer block features on early predictions using Inter CTC residual modules to relax the conditional independence assumption of CTC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' This allows us to close the gap in WER between autoregressive and non-autoregressive meth- ods based on S2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We propose to replace the Efficient Conformer grouped attention by a more efficient and simpler at- tention mechanism that we call patch attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Patch attention reaches similar performance to grouped at- tention with a lower complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We experiment on publicly available LRS2 and LRS3 datasets and reach state-of-the-art results using audio and visual modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Method In this section, we describe our proposed Audio-Visual Efficient Conformer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The model is composed of 4 main components: An audio encoder, a visual encoder, an audio-visual fusion module and an audio-visual encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The audio and visual encoders are separated into modality specific front-end networks to transform each input modal- ity into temporal sequences and Efficient Conformer back- end networks to model local and global temporal relation- ships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The full model is trained end-to-end using intermedi- ate CTC losses between Conformer blocks in addition to the output CTC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The complete architecture of the model is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Model Architecture Audio front-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The audio front-end network first transforms raw audio wave-forms into mel-spectrograms using a short-time Fourier transform computed over win- dows of 20ms with a step size of 10ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 80-dimensional mel-scale log filter banks are applied to the resulting fre- quency features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The mel-spectrograms are processed by a 2D convolution stem to extract local temporal-frequency features, resulting in a 20ms frame rate signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The audio front-end architecture is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 1: Audio Front-end architecture, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 Millions param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Ta denotes the input audio sample length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Stage Layers Output Shape Fourier Transf STFT: 400 window length 160 hop length, 512 ffts (257, Ta//160 + 1) Mel Scale Mel Scale: 80 mels (80, Ta//160 + 1) Stem Conv2d: 32, 180 filters, 22 stride (180, 40, Ta//320 + 1) Proj Linear, 180 units (Ta//320 + 1, 180) Visual front-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The visual front-end network [29] transforms input video frames into temporal sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' A 3D convolution stem with kernel size 5 × 7 × 7 is first ap- plied to the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Each video frame is then processed inde- pendently using a 2D ResNet-18 [20] with an output spatial average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Temporal features are then projected to the back-end network input dimension using a linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The visual front-end architecture is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 2: Visual Front-end architecture, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 Millions pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Tv denotes the number of input video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Stage Layers Output Shape Stem Conv3d: 5 × 72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 64 filters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 1 × 22 stride MaxPoo3d: 1 × 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 1 × 22 stride (64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Tv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 22) Res 1 2 × � Conv2d: 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 64 filters Conv2d: 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 64 filters � (Tv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 22) Res 2 2 × � Conv2d: 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 128 filters Conv2d: 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 128 filters � (Tv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 11) Res 3 2 × � Conv2d: 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 256 filters Conv2d: 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 256 filters � (Tv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 6) Res 4 2 × � Conv2d: 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 512 filters Conv2d: 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 512 filters � (Tv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 3) Pool Global Average Pooling (Tv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 512) Proj Linear,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 256 units (Tv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 256) Back-end networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The back-end networks use an Ef- ficient Conformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The Efficient Conformer encoder was proposed in [7], it is composed of several stages where each stage comprises a number of Conformer blocks [16] using grouped attention with relative positional encodings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The temporal sequence is progressively down- sampled using strided convolutions and projected to wider feature dimensions, lowering the amount of computation while achieving better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We use 3 stages in the audio back-end network to downsample the audio signal to a 80 milliseconds frame rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Only 2 stages are necessary to downsample the visual signal to the same frame rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Ta- ble 6 shows the hyper-parameter of each back-end network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 3: Back-end networks hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' InterCTC blocks indicates Conformer blocks having a post Inter CTC residual module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Network Visual Back-end Audio Back-end Audio-Visual Encoder Num Params (M) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 Num Stages 2 3 1 Blocks per Stage 6, 1 5, 6, 1 5 Total Num Blocks 7 12 5 Stage Feature Dim 256, 360 180, 256, 360 360 Conv Kernel Size 15 15 15 Stage Patch Size 1, 1 3, 1, 1 1 InterCTC Blocks 3, 6 8, 11 2 Audio-visual fusion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Similar to [36, 29], we use an early fusion strategy to learn audio-visual features and reduce model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The acoustic and visual fea- tures from the back-end networks are concatenated and fed into a joint feed-forward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The concatenated fea- tures of size 2 × dmodel are first expanded using a linear layer with output size dff = 4 × dmodel, passed through a Swish activation function [38] and projected back to the original feature dimension dmodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Audio-visual encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The audio-visual encoder is a single stage back-end network composed of 5 Conformer blocks without downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The encoder outputs are then projected to a CTC layer to maximize the sum of prob- abilities of correct target alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Patch Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The Efficient Conformer [7] proposed to replace Multi- Head Self-Attention (MHSA) [44] in earlier encoder lay- ers with grouped attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Grouped MHSA reduce atten- tion complexity by grouping neighbouring temporal ele- ments along the feature dimension before applying scaled dot-product attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Attention having a quadratic com- putational complexity with respect to the sequence length, this caused the network to have an asymmetric complexity with earlier attention layers requiring more flops than latter layers with shorter sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In this work, we pro- pose to replace grouped attention with a simpler and more efficient attention mechanism that we call patch attention (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Similar to the pooling attention proposed by the Multiscale Vision Transformer (MViT) [13] for video and image recognition, the patch attention proceed to an average Table 4: Attention variants complexities including query, key, value and output linear projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' n and d are the sequence length and feature dimension respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Attention Variant Hyper Parameter Full Attention Complexity Regular O(n · d2 + n2 · d) Grouped Group Size (g) O(n · d2 + (n/g)2 · d · g) Patch Patch Size (k) O(n/k · d2 + (n/k)2 · d) AvgPool 1 2 3 4 5 6 7 8 9 a a a b b b c c c Upsample a c Attention b a c b Figure 2: Patch Multi-Head Self-Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The input sequence is downsampled using an average pooling before applying multi-head self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The output sequence is then upsampled via nearest neighbor upsampling, reducing attention com- plexity from O(n2 · d) to O((n/k)2 · d) where k defines the pooling / upsampling kernel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Patch attention is equivalent to regular attention when k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' pooling on the input sequence before projection the query, key and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' X = AvgPooling1d(Xin) (1) with Q, K, V = XW Q, XW K, XW V (2) Where W Q, W K, W V ∈ Rd×d are query, key and value linear projections parameter matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' MHSA with relative sinusoidal positional encoding is then performed at lower resolution as: MHSA(X) = Concat (O1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=', OH) W O (3) with Oh = softmax �QhKT h + Srel h √dh � Vh (4) Where Srel ∈ Rn×n is a relative position score matrix that satisfy Srel[i, j] = QiET j−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' E is the linear projection of a standard sinusoidal positional encoding matrix with posi- tions ranging from −(nmax − 1) to (nmax − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The atten- tion output sequence is then projected and up-sampled back to the initial resolution using nearest neighbor up-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Xout = UpsampleNearest1d(MHSA(X)) (5) In consequence, each temporal element of the same patch produce the same attention output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Local temporal relation- ships are only modeled in the convolution modules while global relationships are modeled by patch attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We use 1-dimensional patches in this work but patch attention Audio back-end Conformer Stage Module Giga FLOPs 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 Stage 1 (d=180, n=500) Stage 2 (d=256, n=250) Stage 3 (d=360, n=125) attention grouped attention (g=3) patch attention (k=3) feed-forward Figure 3: Audio-only back-end modules FLOPs (Billion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' could also be generalized to image and video data using 2D and 3D patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We leave this to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The computational complexity of each attention variant is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Path attention further reduce complexity com- pared to grouped attention by decreasing the amount of computation needed by Query, Key, Value and Output fully connected layers while keeping the feature dimension un- changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Similar to previous work [7], we only use patch attention in the first audio back-end stage to reduce com- plexity while maintaining model recognition performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Figure 3 shows the amount of FLOPs for each attention module variant with respect to encoded sequence length n and model feature dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Using patch or grouped at- tention variants instead of regular MHSA greatly reduce the amount of FLOPs in the first audio back-end stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Intermediate CTC Predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Inspired by [27] and [33] who proposed to add interme- diate CTC losses between encoder blocks to improve CTC- based speech recognition performance, we add Inter CTC residual modules (Figure 4) in encoder networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We con- dition intermediate block features of both audio, visual and audio-visual encoders on early predictions to relax the con- ditional independence assumption of CTC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' During both training and inference, each intermediate prediction is summed to the input of the next layer to help recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We use the same method proposed in [33] except that we do not share layer parameters between losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The lth block output Xout l is passed through a feed-forward network with residual connection and a softmax activation function: Zl = Softmax(Linear(Xout l )) (6) Xin l+1 = Xout l + Linear(Zl) (7) Where Zl ∈ RT ×V is a probability distribution over the output vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The intermediate CTC loss is then com- puted using the target sequence y as: Linter l = −log(P(y|Zl)) (8) with P(y|Zl) = � π∈B−1 CT C(y) T � t=1 Zt,πt (9) Conformer Block Inter CTC Residual Module Conformer Block Linear Softmax Linear CTC loss + Figure 4: Inter CTC residual module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Intermediate pre- dictions are summed to the input of the next Conformer block to condition the prediction of the final block on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Intermediate CTC losses are added to the output CTC loss for the computation of the final loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Where π ∈ V T are paths of tokens and BCT C is a many-to- one map that simply removes all blanks and repeated labels from the paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The total training objective is defined as follows: L = (1 − λ)LCT C + λLinter (10) with Linter = 1 K � k∈interblocks Linter k (11) Where interblocks is the set of blocks having a post Inter CTC residual module (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Similar to [33], we use Inter CTC residual modules every 3 Conformer blocks with λ set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 in every experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Experiments 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Datasets We use 3 publicly available AVSR datasets in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The Lip Reading in the Wild (LRW) [8] dataset is used for visual pre-training and the Lip Reading Sentences 2 (LRS2) [1] and Lip Reading Sentences 3 (LRS3) [2] datasets are used for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' LRW dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' LRW is an audio-visual word recogni- tion dataset consisting of short video segments containing a single word out of a vocabulary of 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The dataset com- prise 488,766 training samples with at least 800 utterances per class and a validation and test sets of 25,000 samples containing 50 utterances per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' LRS2 & LRS3 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The LRS2 dataset is composed of 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 hours with 144,482 videos clips from the BBC tele- vision whereas the LRS3 dataset consists of 438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 hours with 151,819 video clips extracted from TED and TEDx talks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Both datasets include corresponding subtitles with word alignment boundaries and are composed of a pre-train split, train-val split and test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' LRS2 has 96,318 utter- ances for pre-training (195 hours), 45,839 for training (28 hours), 1,082 for validation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 hours), and 1,243 for test- ing (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Whereas LRS3 has 118,516 utterances in the pre-training set (408 hours), 31,982 utterances in the training-validation set (30 hours) and 1,321 utterances in the test set (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' All videos contain a single speaker, have a 224 × 224 pixels resolution and are sampled at 25 fps with 16kHz audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Implementation Details Pre-processing Similar to [29], we remove differences related to rotation and scale by cropping the lip regions us- ing bounding boxes of 96 × 96 pixels to facilitate recog- nition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The RetinaFace [11] face detector and Face Align- ment Network (FAN) [6] are used to detect 68 facial land- marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The cropped images are then converted to gray-scale and normalised between −1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Facial landmarks of the LRW, LRS2 and LRS3 datasets are obtained from previous work [30] and reused for pre-processing to get a clean com- parison of the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' A byte-pair encoding tokenizer is built from LRS2&3 pre-train and trainval splits transcripts using sentencepiece [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We use a vocabulary size of 256 including the CTC blank token following preceding works on CTC-based speech recognition [31, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Data augmentation Spec-Augment [35] is applied on the audio mel-spectrograms during training to prevent over- fitting with two frequency masks with mask size parameter F = 27 and five time masks with adaptive size pS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Similarly to [30], we mask videos on the time axis using one mask per second with the maximum mask duration set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Random cropping with size 88×88 and horizontal flipping are also performed for each video during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We also follow Prajwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [37] using central crop with horizontal flipping at test time for visual-only experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Training Setup We first pre-train the visual encoder on the LRW dataset [8] using cross-entropy loss to recognize words being spoken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The visual encoder is pre-trained for 30 epochs and front-end weights are then used as initializa- tion for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Audio and visual encoders are trained on the LRS2&3 datasets using a Noam schedule [44] with 10k warmup steps and a peak learning rate of 1e-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We use the Adam optimizer [24] with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' L2 regular- ization with a 1e-6 weight is also added to all the trainable weights of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We train all models with a global batch size of 256 on 4 GPUs, using a batch size of 16 per GPU with 4 accumulated steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Nvidia A100 40GB GPUs are used for visual-only and audio-visual experiments while RTX 2080 Ti are used for audio-only experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The audio-only models are trained for 200 epochs while visual- only and audio-visual models are trained for 100 and 70 epochs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Note that we only keep videos shorter than 400 frames (16 seconds) during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Finally, we average models weights over the last 10 epoch checkpoints using Stochastic Weight Averaging [22] before evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 5: Comparison of WER (%) on LRS2 / LRS3 test sets with recently published methods using publicly and non-publicly available datasets for Audio-Only (AO), Visual-Only (VO) and Audio-Visual (AV) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Method Model Criterion Training Datasets Total Hours test WER AO VO AV (↓) Using Publicly Available Datasets (↓) Petridis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [36] CTC+S2S LRW, LRS2 381 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 / - 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 / - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='0 / - Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [48] S2S LRW, LRS2&3 788 / 790 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 / 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 Afouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [3] CTC VoxCeleb2clean, LRS2&3 1,032 / 808 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 / 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [45] S2S LRW, LRS3 595 / 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 / 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 / 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [46] LF-MMI LRS2 224 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 / - 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 / - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 / - Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [29] CTC+S2S LRW, LRS2&3 381 / 595 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 / 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 Prajwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [37] S2S LRS2&3 698 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 / 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [30] CTC+S2S LRW, LRS2&3 818 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 / 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 Ours CTC LRW, LRS2&3 818 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 / 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 + Neural LM CTC LRW, LRS2&3 818 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 / 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 (↓) Using Non-Publicly Available Datasets (↓) Afouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [1] S2S MVLRS, LRS2&3 1,395 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 / 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 / 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 / 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [49] S2S MVLRS, LRS2 954 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 / - Shillingford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [40] CTC LRVSR 3,886 / 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 Makino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [32] Transducer YouTube-31k 31,000 / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 / 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 Serdyuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [39] Transducer YouTube-90k 91,000 / 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 Prajwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [37] S2S MVLRS, TEDxext, LRS2&3 2,676 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 / 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [30] CTC+S2S LRW, AVSpeech, LRS2&3 1,459 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 / 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 Language Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Similarly to [28], we experiment with a N-gram [21] statistical language model (LM) and a Transformer neural language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' A 6-gram LM is used to generate a list of hypotheses using beam search and an external Transformer LM is used to rescore the final list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The 6-gram LM is trained on the LRS2&3 pre-train and train-val transcriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Concerning the neural LM, we pre- train a 12 layer GPT-3 Small [5] on the LibriSpeech LM corpus for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5M steps using a batch size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1M tokens and finetune it for 10 epochs on the LRS2&3 transcriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Results Table 5 compares WERs of our Audio-Visual Effi- cient Conformer with state-of-the-art methods on the LRS2 and LRS3 test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Our Audio-Visual Efficient Con- former achieves state-of-the-art performances with WER of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3%/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' On the visual-only track, our CTC model com- petes with most recent autoregressive methods using S2S criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We were able to recover similar results but still lack behind Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [30] which uses auxiliary losses with pre-trained audio-only and visual-only networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We found our audio-visual network to converge faster than audio-only experiments, reaching better performance using 4 times less training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The intermediate CTC losses of the visual encoder could reach lower levels than in visual-only experi- ments showing that optimizing audio-visual layers can help pre-fusion layers to learn better representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Ablation Studies We propose a detailed ablation study to better understand the improvements in terms of complexity and WER brought by each architectural modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We report the number of operations measured in FLOPs (number of multiply-and- add operations) for the network to process a ten second au- dio/video clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Inverse Real Time Factor (Inv RTF) is also measured on the LRS3 test set by decoding with a batch size 1 on a single Intel Core i7-12700 CPU thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' All abla- tions were performed by training audio-only models for 200 epochs and visual-only / audio-visual models for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Efficient Conformer Visual Back-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We improve the recently proposed visual Conformer encoder [29] using an Efficient Conformer back-end network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The use of byte pair encodings for tokenization instead of characters allows us to further downsample temporal sequences without impacting the computation of CTC loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 6 shows that using an Efficient Conformer back-end network for our visual-only model leads to better performances while reducing model complexity and training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The number of model param- eters is also slightly decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 6: Ablation study on visual back-end network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Visual Back-end #Params (Million) LRS2 test LRS3 test #FLOPs (Billion) Inv RTF Conformer 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='53 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='14 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='94 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='17 Eff Conf 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='39 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='96 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Reference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='the authors looked at papers written over a 10 year period and hundreds had to be thrown out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Outputs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3: the otho looing pa people we over s any your per and conndries that aboutent threghow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6: the autthherss looking paperss we overai year paiod and hundreds that about thrououtow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9: the authors looked at papers witen over ainght year period and hundreds that to been throw out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Block 12: the authors looked at papers written over 10 year period and hundreds had to be thrown out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='Figure 5: Output example of our Visual-only model using greedy search decoding on the LRS3 test set with intermediate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='CTC prediction every 3 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' The sentence is almost correctly transcribed except for the missing ’a’ before ’10 year’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Inter CTC residual modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Similar to [33], we exper- iment adding Inter CTC residual modules between blocks to relax the conditional independence assumption of CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 7 shows that using intermediate CTC losses every 3 Conformer blocks greatly helps to reduce WER, except for the audio-only setting where this does not improve perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Figure 5 gives an example of intermediate block predictions decoded using greedy search without an exter- nal language model on the test set of LRS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We can see that the output is being refined in the encoder layers by con- ditioning on the intermediate predictions of previous lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Since our model refines the output over the frame-level predictions, it can correct insertion and deletion errors in addition to substitution errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We further study the im- pact of Inter CTC on multi-modal learning by measuring the performance of our audio-visual model when one of the two modalities is masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' As pointed out by preced- ing works [8, 1, 32], networks with multi-modal inputs can often be dominated by one of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In our case speech recognition is a significantly easier problem than lip reading which can cause the model to ignore visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Ta- ble 8 shows that Inter CTC can help to counter this problem by forcing pre-fusion layers to transcribe the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 7: Ablation study on Inter CTC residual modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Model Back-end #Params (Million) LRS2 test LRS3 test #FLOPs (Billion) Inv RTF Audio-only (↓) Eff Conf 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='54 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='98 + Inter CTC 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='11 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='67 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='30 Visual-only (↓) Eff Conf 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='39 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='96 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='26 + Inter CTC 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='82 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='63 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='26 Audio-visual (↓) Eff Conf 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='54 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='84 + Inter CTC 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='99 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='82 Table 8: Impact of Inter CTC on audio-visual model WER (%) for LRS2 / LRS3 test sets in a masked modality setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Inter CTC Audio-Visual Eval Mode masked video masked audio no mask No 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='48 / 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='22 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='77 / 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='87 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='54 Yes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='39 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='38 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='62 / 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='58 / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='99 Patch multi-head self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We experiment re- placing grouped attention by patch attention in the first audio encoder stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Our objective being to increase the model efficiency and simplicity without harming perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Grouped attention was proposed in [7] to reduce attention complexity for long sequences in the first encoder stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 9 shows the impact of each attention variant on our audio-only model performance and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We start with an Efficient Conformer (M) [7] and replace the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We find that grouped attention can be replaced by patch attention without a loss of performance using a patch size of 3 in the first back-end stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 9: Ablation study on audio back-end attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Attention Type Group / Patch Size LRS2 test LRS3 test #FLOPs (Billion) Inv RTF Regular 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='66 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='86 Grouped 3, 1, 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='06 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='27 Patch 3, 1, 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='54 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Noise Robustness We measure model noise robustness using various types of noise and compare our Audio-Visual Efficient Conformer with recently published methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Figure 6 shows the WER evolution of audio-only (AO), visual-only (VO) and audio- visual (AV) models with respect to multiple Signal to Noise Ratio (SNR) using white noise and babble noise from the NoiseX corpus [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We find that processing both audio and visual modalities can help to significantly improve speech recognition robustness with respect to babble noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' More- over, we also experiment adding babble noise during train- ing as done in previous works [36, 29] and find that it can further improve noise robustness at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Robustness to various types of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We gather var- ious types of recorded audio noise including sounds and music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Table 10, we observe that the Audio-Visual Ef- ficient Conformer consistently achieves better performance than its audio-only counterpart in the presence of various noise types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' This confirm our hypothesis that the audio- visual model is able to use the visual modality to aid speech recognition when audio noise is present in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' SNR (dB) Word Error Rate (%) 0 10 20 30 40 50 5 0 5 10 15 20 VO LRS2 AO LRS2 AV LRS2 AV* LRS2 VO LRS3 AO LRS3 AV LRS3 AV* LRS3 (a) Babble noise SNR (dB) Word Error Rate (%) 0 10 20 30 40 50 5 0 5 10 15 20 VO LRS2 AO LRS2 AV LRS2 AV* LRS2 VO LRS3 AO LRS3 AV LRS3 AV* LRS3 (b) White noise Figure 6: LRS2 and LRS3 test WER (%) as a function of SNR (dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' * indicates experiments being trained with babble noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We measure noise robustness by evaluating our models in the presence of babble and white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 10: LRS3 test WER (%) as a function of SNR (dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Noise Mode SNR (dB) 5 0 5 10 15 20 babble AO 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 AV* 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 train AO 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 AV 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 AV* 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 Comparison with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We compare our method with results provided by Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [29] and Petridis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [36] on the LRS2 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 11 shows that our audio-visual model achieves lower WER in the pres- ence of babble noise, reaching WER of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7% at -5 dB SNR against 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3% for Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Table 11: Comparison with Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' LRS2 test WER (%) as a function of SNR (dB) using babble noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Method Mode SNR (dB) 5 0 5 10 15 20 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [29] VO 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 AO* 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 AV* 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 Ours VO 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 AO 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 27 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 AV 25 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 AV* 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 Table 12: Comparison with Petridis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' LRS2 test WER (%) as a function of SNR (dB) using white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Method Mode SNR (dB) 5 0 5 10 15 20 Petridis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [36] VO 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 AO* 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 AV* 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 Ours VO 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6 AO 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 AV 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 AV* 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Conclusion In this paper, we proposed to improve the noise robust- ness of the recently proposed Efficient Conformer CTC- based architecture by processing both audio and visual modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We showed that incorporating multi-scale CTC losses between blocks could help to improve recognition performance, reaching comparable results to most recent autoregressive lip reading methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We also proposed patch attention, a simpler and more efficient attention mechanism to replace grouped attention in the first audio encoder stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Our Audio-Visual Efficient Conformer achieves state-of- the-art performance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='3% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='8% on the LRS2 and LRS3 test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In the future, we would like to explore other techniques to further improve the noise robustness of our model and close the gap between recent lip reading methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' This includes adding various audio noises during training and using cross-modal distillation with pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' We also wish to reduce the visual front-end net- work complexity without arming recognition performance and experiment with the RNN-Transducer learning objec- tive for streaming applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Acknowledgments This work was partly supported by The Alexander von Humboldt Foundation (AvH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' References [1] Triantafyllos Afouras, Joon Son Chung, Andrew Senior, Oriol Vinyals, and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Deep audio-visual speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [2] Triantafyllos Afouras, Joon Son Chung, and Andrew Zisser- man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Lrs3-ted: a large-scale dataset for visual speech recog- nition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='00496, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [3] Triantafyllos Afouras, Joon Son Chung, and Andrew Zisser- man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Asr is all you need: Cross-modal distillation for lip reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In ICASSP 2020-2020 IEEE International Confer- ence on Acoustics, Speech and Signal Processing (ICASSP), pages 2143–2147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [4] Yannis M Assael, Brendan Shillingford, Shimon Whiteson, and Nando De Freitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Lipnet: End-to-end sentence-level lipreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='01599, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [5] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Sub- biah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakan- tan, Pranav Shyam, Girish Sastry, Amanda Askell, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Lan- guage models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Advances in neural in- formation processing systems, 33:1877–1901, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [6] Adrian Bulat and Georgios Tzimiropoulos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' How far are we from solving the 2d & 3d face alignment problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' (and a dataset of 230,000 3d facial landmarks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceedings of the IEEE International Conference on Computer Vision, pages 1021–1030, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [7] Maxime Burchi and Valentin Vielzeuf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Efficient conformer: Progressive downsampling and grouped attention for auto- matic speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In 2021 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU), pages 8– 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [8] Joon Son Chung and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Lip reading in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Asian conference on computer vision, pages 87–103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [9] Joon Son Chung and AP Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Lip reading in profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [10] Ronan Collobert, Christian Puhrsch, and Gabriel Synnaeve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Wav2letter: an end-to-end convnet-based speech recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='03193, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [11] Jiankang Deng, Jia Guo, Evangelos Ververas, Irene Kot- sia, and Stefanos Zafeiriou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Retinaface: Single-shot multi- level face localisation in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 5203–5212, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [12] Linhao Dong, Shuang Xu, and Bo Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Speech-transformer: a no-recurrence sequence-to-sequence model for speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 5884–5888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [13] Haoqi Fan, Bo Xiong, Karttikeya Mangalam, Yanghao Li, Zhicheng Yan, Jitendra Malik, and Christoph Feichten- hofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Multiscale vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vi- sion, pages 6824–6835, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [14] Alex Graves, Santiago Fern´andez, Faustino Gomez, and J¨urgen Schmidhuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In ICML, pages 369–376, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [15] Alex Graves, Abdel-rahman Mohamed, and Geoffrey Hin- ton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Speech recognition with deep recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In 2013 IEEE international conference on acoustics, speech and signal processing, pages 6645–6649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Ieee, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [16] Anmol Gulati, James Qin, Chung-Cheng Chiu, Niki Par- mar, Yu Zhang, Jiahui Yu, Wei Han, Shibo Wang, Zheng- dong Zhang, Yonghui Wu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Conformer: Convolution- augmented transformer for speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='08100, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [17] Pengcheng Guo, Florian Boyer, Xuankai Chang, Tomoki Hayashi, Yosuke Higuchi, Hirofumi Inaguma, Naoyuki Kamo, Chenda Li, Daniel Garcia-Romero, Jiatong Shi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Recent developments on espnet toolkit boosted by con- former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In ICASSP 2021-2021 IEEE International Confer- ence on Acoustics, Speech and Signal Processing (ICASSP), pages 5874–5878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [18] Wei Han, Zhengdong Zhang, Yu Zhang, Jiahui Yu, Chung- Cheng Chiu, James Qin, Anmol Gulati, Ruoming Pang, and Yonghui Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Contextnet: Improving convolutional neural networks for automatic speech recognition with global con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='03191, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [19] Awni Hannun, Carl Case, Jared Casper, Bryan Catanzaro, Greg Diamos, Erich Elsen, Ryan Prenger, Sanjeev Satheesh, Shubho Sengupta, Adam Coates, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Deep speech: Scaling up end-to-end speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='5567, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [20] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceed- ings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [21] Kenneth Heafield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Kenlm: Faster and smaller language model queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceedings of the sixth workshop on sta- tistical machine translation, pages 187–197, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [22] Pavel Izmailov, Dmitrii Podoprikhin, Timur Garipov, Dmitry Vetrov, and Andrew Gordon Wilson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Averaging weights leads to wider optima and better generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='05407, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [23] Shigeki Karita, Nanxin Chen, Tomoki Hayashi, Takaaki Hori, Hirofumi Inaguma, Ziyan Jiang, Masao Someki, Nelson Enrique Yalta Soplin, Ryuichi Yamamoto, Xiaofei Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' A comparative study on transformer vs rnn in speech applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In 2019 IEEE Automatic Speech Recog- nition and Understanding Workshop (ASRU), pages 449– 456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [24] Diederik P Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [25] Samuel Kriman, Stanislav Beliaev, Boris Ginsburg, Joce- lyn Huang, Oleksii Kuchaiev, Vitaly Lavrukhin, Ryan Leary, Jason Li, and Yang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Quartznet: Deep automatic speech recognition with 1d time-channel separable convolu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 6124–6128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [26] Taku Kudo and John Richardson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Sentencepiece: A sim- ple and language independent subword tokenizer and detok- enizer for neural text processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In EMNLP, pages 66–71, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [27] Jaesong Lee and Shinji Watanabe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Intermediate loss regular- ization for ctc-based speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In ICASSP 2021- 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 6224–6228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [28] Jason Li, Vitaly Lavrukhin, Boris Ginsburg, Ryan Leary, Oleksii Kuchaiev, Jonathan M Cohen, Huyen Nguyen, and Ravi Teja Gadde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Jasper: An end-to-end convolutional neu- ral acoustic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='03288, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [29] Pingchuan Ma, Stavros Petridis, and Maja Pantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' End- to-end audio-visual speech recognition with conformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 7613–7617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [30] Pingchuan Ma, Stavros Petridis, and Maja Pantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Visual speech recognition for multiple languages in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='13084, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [31] Somshubra Majumdar, Jagadeesh Balam, Oleksii Hrinchuk, Vitaly Lavrukhin, Vahid Noroozi, and Boris Ginsburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Cit- rinet: Closing the gap between non-autoregressive and au- toregressive end-to-end models for automatic speech recog- nition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='01721, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [32] Takaki Makino, Hank Liao, Yannis Assael, Brendan Shillingford, Basilio Garcia, Otavio Braga, and Olivier Sio- han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Recurrent neural network transducer for audio-visual speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In 2019 IEEE automatic speech recog- nition and understanding workshop (ASRU), pages 905–912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [33] Jumon Nozaki and Tatsuya Komatsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Relaxing the con- ditional independence assumption of ctc-based asr by con- ditioning on intermediate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='02724, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [34] Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev Khudanpur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Librispeech: an asr corpus based on public do- main audio books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In 2015 IEEE international conference on acoustics, speech and signal processing (ICASSP), pages 5206–5210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [35] Daniel S Park, Yu Zhang, Chung-Cheng Chiu, Youzheng Chen, Bo Li, William Chan, Quoc V Le, and Yonghui Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Specaugment on large scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In ICASSP, pages 6879–6883, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [36] Stavros Petridis, Themos Stafylakis, Pingchuan Ma, Geor- gios Tzimiropoulos, and Maja Pantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Audio-visual speech recognition with a hybrid ctc/attention architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In 2018 IEEE Spoken Language Technology Workshop (SLT), pages 513–520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [37] KR Prajwal, Triantafyllos Afouras, and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Sub-word level lip reading with visual attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceed- ings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5162–5172, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [38] Prajit Ramachandran, Barret Zoph, and Quoc V Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Searching for activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='05941, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [39] Dmitriy Serdyuk, Otavio Braga, and Olivier Siohan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Audio- visual speech recognition is worth 32x32x8 voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In 2021 IEEE Automatic Speech Recognition and Understand- ing Workshop (ASRU), pages 796–802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [40] Brendan Shillingford, Yannis Assael, Matthew W Hoff- man, Thomas Paine, C´ıan Hughes, Utsav Prabhu, Hank Liao, Hasim Sak, Kanishka Rao, Lorrayne Bennett, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Large-scale visual speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' arXiv preprint arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content='05162, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [41] Joon Son Chung, Andrew Senior, Oriol Vinyals, and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Lip reading sentences in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceed- ings of the IEEE conference on computer vision and pattern recognition, pages 6447–6456, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [42] George Sterpu, Christian Saam, and Naomi Harte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Attention- based audio-visual fusion for robust automatic speech recog- nition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceedings of the 20th ACM International Con- ference on Multimodal Interaction, pages 111–115, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [43] Andrew Varga and Herman JM Steeneken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Assessment for automatic speech recognition: Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' noisex-92: A database and an experiment to study the effect of additive noise on speech recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Speech communication, 12(3):247– 251, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [44] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [45] Bo Xu, Cheng Lu, Yandong Guo, and Jacob Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Discrim- inative multi-modality speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 14433–14442, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [46] Jianwei Yu, Shi-Xiong Zhang, Jian Wu, Shahram Ghorbani, Bo Wu, Shiyin Kang, Shansong Liu, Xunying Liu, Helen Meng, and Dong Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Audio-visual recognition of overlapped speech for the lrs2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In ICASSP 2020-2020 IEEE In- ternational Conference on Acoustics, Speech and Signal Pro- cessing (ICASSP), pages 6984–6988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [47] Qian Zhang, Han Lu, Hasim Sak, Anshuman Tripathi, Erik McDermott, Stephen Koo, and Shankar Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Transformer transducer: A streamable speech recognition model with transformer encoders and rnn-t loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 7829–7833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [48] Xingxuan Zhang, Feng Cheng, and Shilin Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Spatio- temporal fusion based convolutional sequence learning for lip reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 713–722, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' [49] Ya Zhao, Rui Xu, Xinchao Wang, Peng Hou, Haihong Tang, and Mingli Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' Hearing lips: Improving lip reading by dis- tilling speech recognizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} +page_content=' In Proceedings of the AAAI Con- ference on Artificial Intelligence, volume 34, pages 6917– 6924, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQffvxD/content/2301.01456v1.pdf'} diff --git a/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf b/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6b1b917eff7728863b0bccf32bba2b9c8b17271c --- /dev/null +++ b/7tAyT4oBgHgl3EQf2_kh/content/2301.00759v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d7cb062df0dd7a524b9318b32f476a4edc0e8ac2df34ee081a8feff2f7af3da3 +size 3251879 diff --git a/7tAyT4oBgHgl3EQf2_kh/vector_store/index.faiss b/7tAyT4oBgHgl3EQf2_kh/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5d8e50c4b5ae85ec94f79a966e7cff89d534e5c7 --- /dev/null +++ b/7tAyT4oBgHgl3EQf2_kh/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d14c9a80467d6115ec1b86d4d670b702f47deb1e129cb11c4a2302ae2d626e54 +size 4587565 diff --git a/7tAyT4oBgHgl3EQf2_kh/vector_store/index.pkl b/7tAyT4oBgHgl3EQf2_kh/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..4d6a199221f31efe9548de4334af5647cfbc764b --- /dev/null +++ b/7tAyT4oBgHgl3EQf2_kh/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d785fa621d073574f78a94ab19df8174a3ae3f9fa8cd65430b96dd7fdede3e0f +size 171099 diff --git a/7tE1T4oBgHgl3EQfTwM_/content/tmp_files/2301.03081v1.pdf.txt b/7tE1T4oBgHgl3EQfTwM_/content/tmp_files/2301.03081v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d92db57e9b1871b2a1eab52886d80ebbf08d3bd9 --- /dev/null +++ b/7tE1T4oBgHgl3EQfTwM_/content/tmp_files/2301.03081v1.pdf.txt @@ -0,0 +1,1214 @@ + +1 + +Abstract— Objective: The objective of this study is to develop a +deep-learning based detection and diagnosis technique for carotid +atherosclerosis using a portable freehand 3D ultrasound (US) +imaging system. Methods: A total of 127 3D carotid artery datasets +were acquired using a portable 3D US imaging system. A U-Net +segmentation network was firstly applied to extract the carotid +artery on 2D transverse frame, then a novel 3D reconstruction +algorithm using fast dot projection (FDP) method with position +regularization was proposed to reconstruct the carotid artery +volume. Furthermore, a convolutional neural network was used +to classify the healthy case and diseased case qualitatively. 3D +volume analysis including longitudinal reprojection algorithm and +stenosis grade measurement algorithm was developed to obtain the +clinical metrics quantitatively. Results: The proposed system +achieved sensitivity of 0.714, specificity of 0.851 and accuracy of +0.803 respectively in diagnosis of carotid atherosclerosis. The +automatically +measured +stenosis +grade +illustrated +good +correlation (r=0.762) with the experienced expert measurement. +Conclusion: the developed technique based on 3D US imaging can +be applied to the automatic diagnosis of carotid atherosclerosis. +Significance: The proposed deep-learning based technique was +specially designed for a portable 3D freehand US system, which +can +provide +carotid +atherosclerosis +examination +more +conveniently and decrease the dependence on clinician’s +experience. +Index Terms—3D ultrasound imaging, automatic carotid +atherosclerosis +diagnosis, +carotid +artery +segmentation, +reconstruction with regularization. +I. INTRODUCTION +AROTID atherosclerosis is one of the major causes of +stroke which is the world’s second leading cause of death +[1]. The prevalence rate of carotid atherosclerosis is 36.2% in +Chinese people over 40 years old [2]. The pathological features +of carotid atherosclerosis are increase of intima-media +thickness and appearance of atherosclerosis plaque. Magnetic +resonance imaging (MRI), computed tomography angiography + +This work was sponsored by Natural Science Foundation of China (NSFC) +under Grant No.12074258. (Jiawen Li and Yunqian Huang are co-first authors.) +(Corresponding authors: Rui Zheng, Man Chen.) +Jiawen Li, Sheng Song, Duo Xu and Haibin Zhang are with School of +Information Science and Technology, ShanghaiTech University, Shanghai, +China. +Hongbo Chen is with School of Information Science and Technology, +ShanghaiTech University, Shanghai 201210, China, also with Shanghai +Advanced Research Institute, Chinese Academy of Sciences, Shanghai 200050, +(CTA) and digital subtraction angiography (DSA) are several +commonly used methods for visualizing and characterizing +carotid artery features [3]–[5]. However, these methods still +have some limitations during application due to invasiveness, +ionizing radiation, heavy equipment etc.; and the approaches +are very time-consuming and expensive which can’t satisfy the +need of large scale of examinations in different environments +especially for community and countryside areas. 2D Ultrasound +(US), as a non-invasive and low-cost method, is widely used in +the examination of carotid plaque. However, there are several +disadvantages of traditional 2D US in the current ultrasound +examination of carotid atherosclerosis. (1) It is mainly carried +out by experienced sonographers in hospital, and becomes a +huge burden for health care system. (2) Routine health check is +difficult for carotid atherosclerosis patients especially in rural +or undeveloped area. (3) Routine ultrasound examination is a +tedious, +laborious, +experience-dependent +work +for +sonographers. (4) Clinically, some metrics such as intima- +media thickness (IMT), plaque thickness, plaque area, usually +assess the severity of the carotid atherosclerosis in 2D US +images, which is prone to variability and lack of 3D +morphology of carotid plaque [6], [7]. 3D US carotid artery +imaging approaches mainly include mechanical scanning and +tracked freehand scanning using various sensors e.g., magnetic +tracked senor, optical tracked sensor, etc., [8] which can +provide plaque volume estimation, 3D morphology of plaque +and other 3D metrics for carotid atherosclerosis diagnosis. The +3D techniques are found to be more accurate to evaluate the +progress of carotid atherosclerosis [9]–[12]. Therefore, it is of +great importance to develop a portable, reliable and cost- +effective automatic ultrasound diagnostic technique for carotid +atherosclerosis screening. + The automatic diagnosis of carotid atherosclerosis focuses on +finding the biomarkers on the ultrasound images, for example +China, and also with University of Chinese Academy of Sciences, Beijing +100049, China. +Yunqian Huang and Junni Shi are with Tongren Hospital, Shanghai Jiao +Tong University School of Medicine, Shanghai, China. +Man Chen is with Tongren Hospital, Shanghai Jiao Tong University School +of Medicine, Shanghai, China (e-mail: maggiech1221@126.com) +Dr. Rui Zheng is with School of Information Science and Technology, +Shanghai Engineering Research Center of Energy Efficient and Custom AI IC, +ShanghaiTech University, Shanghai, China (phone: 86 21-2068 4452, e-mail: +zhengrui@shanghaitech.edu.cn) +Automatic Diagnosis of Carotid Atherosclerosis +Using a Portable Freehand 3D Ultrasound +Imaging System +Jiawen Li, Yunqian Huang, Sheng Song, Hongbo Chen, Junni Shi, Duo Xu, Haibin Zhang, Man +Chen*, Rui Zheng* +C + + +2 +vessel wall area, vessel wall volume or total plaque volume +[13]–[15]. These biomarkers are all bounded by the two +boundaries of vessels, the media-adventitia boundary (MAB) +and the lumen-intima boundary (LIB), thus identifying these +two boundaries is an important issue during the carotid +atherosclerosis diagnosis. In recent years, deep learning +methods has achieved excellent performance in medical image +processing. Jiang et al. [16]–[18]designed a novel adaptive +triple loss for carotid artery segmentation. To utilize 3D +information in 3D volume of carotid artery, Jiang et al. [19] +introduced a fusion module to the U-Net segmentation network +and yielded promising performance on carotid segmentation +task. Zhou et al.[20] proposed a deep learning-based MAB and +LIB segmentation method, and a dynamic convolutional neural +network (CNN) were applied to image patches in every slice of +the 3D US images. LIB segmentation was performed by U-Net +based on the masks of the MAB since the LIB is inside the +MAB. The method achieved high accuracy but initial anchor +points were still manually placed. Ruijter et al. [21] created a +generalized method to segment LIB using CNN. Several U- +Nets were compared and the experiments showed that the +combination of various vessels such as radial, ulnar artery, or +cephalic vein improved the segmentation performance of +carotid artery. After segmentation, a 3D-geometry can be +obtained for further therapy. Van Knippenberg et al [22] +proposed an unsupervised learning method to solve the lack of +data in carotid segmentation task. Azzopardi et al. [23] +designed a novel geometrically constrained loss functions and +received improved segmentation results. Zhou et al.[24] +proposed a voxel based 3D segmentation neural network to +segment the MAB and LIB in 3D volume directly. Although the +proposed algorithm achieved high accuracy with fast process, +user’s interaction is yet required to identify ROI in the first and +last slice of the volume. +After region of interest (ROI) i.e., carotid artery is identified, +further analysis needs to be performed to get significant clinical +information for carotid atherosclerosis diagnosis such as the +existence of plaque, carotid stenosis grade, type of the plaque, +etc. Zhou et al.[25],[26] applied 8 different backbone and +UNet++ segmentation algorithm trained on 2D longitudinal US +images to segment the plaque region and calculate the total +plaque area. Xia et al. [27] employed a CNN to categorize +segmented carotid images into normal cases, thickening vessel +wall cases and plaque cases. Ma et al.[28] proposed a multilevel +strip pooling-based convolutional neural network to investigate +the echogenicity of plaque which was found to be closely +correlated with the risk of stroke. Shen et al. [29] proposed a +multi task learning method, the authors combined ultrasound +reports and plaque type label to train a CNN to classify four +different plaque type. Zhao et al. [30] utilized a novel vessel +wall thickness mapping algorithm to evaluate the therapeutical +performance on carotid atherosclerosis. Zhou et al. [31] utilized +the unsupervised pretrained parameters of U-Net to train a +plaque segmentation network with a small 3D carotid artery +ultrasound dataset. Saba et al. [32] used a deep learning based +method to measure the carotid stenosis, three deep learning +based systems were evaluated on 407 US dataset, and achieved +AUC of 0.90, 0.94 and 0.86 on the longitudinal US images +respectively. Biswas et al. [33] proposed a two-stage artificial +intelligence model for jointly measurement of atherosclerotic +wall thickness and plaque burden in longitudinal US images. +The results showed that the proposed method achieved the +lowest error compared to previous method. +The current 3D carotid imaging device was mainly based +on mechanical system and hard to transport which was almost +impossible to apply in community or rural area, therefore the +portable freehand 3D ultrasound imaging system was required +which can be easily applied for various scenarios. However, for +the freehand 3D ultrasound reconstruction, the requested small +voxel size and various noise would lead to reconstruction +artifacts[34], [35]. On the other hand, the clinicians in different +scenarios were usually inexperienced so that the diagnosis +results might be inaccurate and hard to reproduce compared +with sonographers in clinical ultrasound department. In this +paper, we developed a new detection and classification +technique based on deep-learning algorithms for carotid +atherosclerosis diagnosis which can be employed to a portable +freehand 3D US imaging system for fast screening. Compared +to other 3D ultrasound carotid artery imaging methods mainly +focusing on carotid vessel wall segmentation [18], [20], [21], +[24], the proposed method aimed at exploring an automatic and +experience-independent technique and framework for fast +carotid arteriosclerosis diagnosis. +The main contributions are outlined as follows. Firstly, a +portable freehand 3D US carotid imaging and diagnosis +framework including deep-learning based segmentation, 3D +reconstruction and automatic volume analysis was developed +for fast carotid atherosclerosis diagnosis. Secondly, a novel +position regularization algorithm was designed to reduce the +reconstruction error caused by freehand scan. Lastly, post +analysis including automatic reprojection and stenosis +measurement from 3D volume data provided visible qualitative +results and quantitative results for atherosclerosis diagnosis. +II. METHODS +Fig. 1 showed the overview of data processing procedure +including transverse image segmentation, 3D volume +reconstruction, detection of carotid atherosclerosis and 3D +carotid volume analysis. +A. MAB and LIB Segmentation +Three consecutive frames were concatenated in channel +dimension which is proved to be useful to improve the +segmentation accuracy [36]. +Since the adjacent frames contained lots of redundant +information, the pseudo labels were generated using pseudo- +labeling method to reduce the work load [37]. One of every 5 +neighbor frames were selected to be manually labeled by +experienced sonographers and the other four frames were +inferred by the network which was trained using the labeled +frames. All generated pseudo labels were checked visually, the +labels would be corrected if the segmentation is incorrect. +The intensity of the image was normalized to [0,1] as follows: +𝐼 = +𝐼 − 𝐼𝑚𝑖𝑛 +𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛 + +(1) + + +3 +where I represented the intensity of the image. Imax and Imin +represent the max and minimum value of the intensity in the US +image. All images and corresponding labels were resized to +224*224 for segmentation network training. +U-Net was employed to segment the MAB and LIB in the +transverse US image sequence [38]. The architecture of the +network was illustrated in Fig. 2. The segmentation module +consisted of two symmetrical sub-module which were encoder +and decoder. The number of channels for each convolutional +layer were set to (64, 128, 256, 512, 512). Each convolutional +layer was followed by a batch normalization module and a +rectification linear unit (ReLU) module. The two modules were +connected using skip connection to exploit all resolution +features. The loss function of the segmentation module was the +combination of DSC loss and cross-entropy loss: +𝐿𝑜𝑠𝑠 = 𝐿𝑜𝑠𝑠𝑑𝑖𝑐𝑒 + 𝐿𝑜𝑠𝑠𝑐𝑒 +(2) +B. 3D Reconstruction with Regularization +After the MAB and LIB were identified in every slice of US +image sequence, the 3D carotid artery volume was +reconstructed using the Fast Dot Projection (FDP) method [39]. +However, some disturbances caused by the low precision of the +magnetic sensor, inevitable hand shaking and breathing +movement during carotid swept, would lead to the +reconstruction errors and artifacts. The major problem was the +repeated acquisition at the same or very close positions, and it +caused large uncertainty at volume voxels and discontinuity in +the reconstructed volume [40]. To improve the image quality +and decrease the uncertainty of 3D reconstructed volume, a +total variation regularization [41] method was integrated with +FDP reconstruction algorithm. +(1) For all the position information obtained from 3DUS +device, it could be formulated as a set of rotation matrix 𝑅 and +a translation 𝑡. The tuple (𝑹, 𝒕) consisting of all 𝑅 and 𝑡 formed +the special Euclidean group 𝑆𝐸(3) which was the semi-direct +product of the rotation group 𝑆𝑂(3) and the translation group. +Therefore, the 𝑆𝐸(3) can be formulated as: +𝑆𝐸(3) = {(𝑅 𝑡 +0 1) : 𝑅 ∈ 𝑆𝑂(3), 𝑡 ∈ ℝ3} +(3) + +Fig. 2. The architecture of the segmentation module. + +Fig. 1. The pipeline of the proposed system and corresponding algorithm. The top row demonstrated the process of the data acquisition, extraction of ROI and +3D reconstruction. The bottom row represented the process of 3D carotid volume analysis. The original image sequence and corresponding position +information were firstly obtained by the 3D US device. U-Net segmentation algorithm and regularized Fast-Dot Projection algorithm was applied to extract +the ROI and 3D carotid volume. Then 3D carotid volume analysis included automatic stenosis grade measurement, longitudinal image reprojection and +healthy/diseased cases classification was conducted based on the reconstructed volume. + + +:Conv+Batchnorm+ReLU × 2 +:Maxpooling +: Upsampling +:Concatenate +:1×1 ConvSegmented masks +Data collection +Pre-processed images +of LIB and MAB +Image +Segmentation +Fast-Dot Projection +3D +Sequence +Reconstruction +Reconstruction +Algorithm +Total +Position +Variation +Information +Fast-Dot Projection +Regularization +3D +Reconstruction +Algorithm +Reconstruction +Stenosis Grade +Stenosis rate +-30° +Measurement +Projection +Reprojection +0° +Module +Projection +Exist Plaque +Diagnosis +or Not ++30° +Projection +4 + (2) The position signal obtained by the 3DUS system could +be considered as a set of entries which forms a vector 𝑷 = +(𝒑𝟏, 𝒑𝟐, … , 𝒑𝒌) ∈ 𝑴𝒌, where 𝑘 was the number of entries and +𝑘 ∈ 𝑁, 𝑀𝑘 was a manifold and 𝑀 = 𝑆𝐸(3). Another signal +set X were considered to be found when the following formula +is minimal. +E(𝐱) = D(𝐱, 𝐩) + αR(𝐱), α > 0 +(4) +Where 𝐷(𝑥, 𝑝) was the penalizing term to reduce the variation +between original signal P and resulted signal X. 𝑅(𝒙) was a +regularized term to penalize the position saltation in the signal +X. + (3) The deviation penalized term D(x,p) could be defined as: +𝐷(𝐱, 𝐟) = ∑   +𝑘 +𝑖=1 +(ℎ ∘ 𝑑)(𝐱𝑖, 𝐩𝑖) +(5) +Where d(xi,pi) was the length of the geodesic which was +defined as a shortest path on 𝑀 between two pose p and q [41]. +ℎ was defined as following: +ℎ(𝑠) = {𝑠2, +𝑠 < 1/√2 +√2𝑠 − 1/2, otherwise +(6) +Which was the Huber-Norm. + (4) For the regularized term, it could be defined as the +following: +𝑅(𝐱) = ∑   +𝑘−1 +𝑖=1 +(ℎ ∘ 𝑑)(𝐱𝑖, 𝐱𝑖+1) +(7) +Where d(xi,xi+1) could be considered as the first-order forward +difference. The optimize problem in (4) could be solved using +a cyclic proximal point algorithm. +However, the original regularized algorithm couldn’t handle +the scanning positions with large backward movements. In this +case, the position array was not sequential according to the +coordinates, therefore pose re-rank algorithm was proposed. +Concretely, considering the centroid point of every frame from +the 2D segmented image sequence as 𝑪𝒌 = (𝒄𝟏, 𝒄𝟐, … , 𝒄𝒌) , the +PCA (principal components analysis) algorithm was conducted +in 𝐶𝑘 and a new matrix 𝐷𝑘 was obtained. The first column of +the matrix was the principal vector 𝑣𝑘, then a set of vectors 𝑐𝑑 +could be acquired by projecting every centroid vector 𝑐𝑘 to 𝑣𝑘. +𝒄𝒅 = 𝒄𝑘 − 𝒄𝒌 ⋅ 𝑣𝑘 +𝑣𝑘 ⋅ 𝑣𝑘 +𝑣𝑘 +(8) +The new position sequence was finally obtained by sorting the +l2-norm of the 𝒄𝒅 set. +C. Carotid Atherosclerosis Diagnosis +The US scans including the segmented and reconstructed +volume were classified into healthy case and carotid +atherosclerosis case using a diagnosis network. As illustrated in +Fig. 3, there were two inputs for the diagnosis module. It had +been proved that the morphological information was helpful for +the network to classify the normal or abnormal (diseased) image +[42], therefore the mask of LIB and MAB extracted from each +slice of the reconstructed volume was used as one input. The +other input was the cropped ROI which was determined by the +max bounding rectangular of the mask, and in the cropped +image, the intensity in the region between LIB and MAB was +set to the original value, while the region inside lumen and +outside vessel wall were set to 0. For each input stream, it +consisted of three repeated blocks, each block consisted of two +consequent basic convolutional sub-block and a max pooling +layer. The basic convolutional sub-block was composed of a +convolutional layer, a batch normalization module and a linear +rectification unit. The number of channels for each repeated +block were set to (24, 48, 96). The fusion block concatenated +the high-level features of two streams and combined +information by introducing a basic convolutional sub-block. +After fusion block, the remaining layers were global average +pooling (GAP) layers and a fully connected layer to output the +diagnosis result. We used focal loss in the diagnosis module. +The scan would be diagnosed as a carotid atherosclerosis +case if the consecutive 5 transverse slices from the +reconstructed volume were classified as existing plaque. +D. 3D Carotid Volume Analysis + The clinical diagnostic parameters such as plaque thickness, +plaque length, stenosis grade, plaque area, plaque type, etc. can +be directly calculated from the reconstructed carotid artery +volume. To validate accuracy of the proposed method, the +longitudinal US images of carotid artery were obtained by +projecting the volume in different angles, and the stenosis grade +was calculated. +Stenosis rate was usually used to evaluate the stenosis grade. +For the slices which were diagnosed as atherosclerosis, stenosis +degree can be evaluated using the LIB and MAB masks. The +diameter stenosis rate was usually calculated to evaluate the +stenosis grade in clinic. We denote it as +𝑆𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 = +𝐿𝑤𝑎𝑙𝑙 +𝐿𝑤𝑎𝑙𝑙 + 𝐿𝑙𝑢𝑚𝑒𝑛 +(9) +where 𝐿 represented the length of respective area. The metric +was ranged from 0 to 1, the greater number indicated the more +severe stenosis. The length of vessel wall 𝐿𝑤𝑎𝑙𝑙 and length of + +Fig. 3. The architecture of the diagnosis module. + +Fig. 4 The illustration of the approach to calculate the diameter stenosis. + +24X128X128 +128X128 +48X64X64 +96X32X32 +96X16X16 +96X16X16 +Exist +plaque +24X128X128 +48X64X64 +or not +96X32X32 +:Conv+Batchnorm+ReLU +96X16X16 +:Maxpooling +128X128 +:Global average pooling +:Concatenate ++ :Fully connectRadially Sampling +:lumen +:Vessel Wall +5 +lumen 𝐿𝑙𝑢𝑚𝑒𝑛 were illustrated as Fig. 4. The diameter stenosis +rate was the max 𝑆𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 which was calculated using all +points in MAB boundary. +The longitudinal carotid US images were usually used to +calculate plaque size and evaluate the morphology of plaque. +Since the carotid artery is curved volume, the direct projection +along a fixed axis may lead to missing of some structure. +Therefore, centroid points of carotid artery in transverse slices +were selected to determine the projection plane. Specifically, as +illustrated in Fig. 5, denoting the centroid point of i-th slice in +the volume as 𝐶𝑖, the line which was 𝜃 degree angled with y- +axis through the centroid point 𝐶𝑖 , was sampled as the i-th +column of projected longitude image. In our experiment, the +longitudinal US images were obtained by reprojecting the 3D +carotid volume at the angles of 0°, ±15° and ±30°. +III. EXPERIMENTAL SETUP +A. Data Acquisition and 3D Ultrasound Scan +A portable freehand 3D ultrasound imaging system was used +to obtain three-dimensional images of carotid artery as shown +in Fig. 6. The system consisted of a 2D linear probe (Clarius, +L738-K, Canada), an electromagnetic tracking system +(Polhemus, G4 unit, U.S.A) and a host laptop computer (Intel +i7-8700k CPU @ 3.70GHz, 32GB RAM) [43]. The 2D +transverse US images were acquired by the probe while the +corresponding position and orientation information were +captured by the magnetic sensor. The images and orientation +were acquired with a frame rate of 24 Hz. +During the acquisition, the subjects took the supine position +and was scanned as shown in Fig. 6 (d), and the probe swept +consistently along the long axis of common carotid artery from +the proximal end to the distal end at the speed of approximate +10-15 seconds per scan. To reduce the reconstruction artifacts, +fallback along the swept direction and large movement normal +to the swept direction should be avoided. The inclusion criteria +were based on visible plaques which was identified by expert. +The stenosis grade larger than 70% was excluded to the dataset. +A total of 127 3D carotid artery scans from 83 subjects with +stenosis range from 0% to 70% were obtained from local +hospital, and all subjects consented to participate in this +experiment, which was approved by the local ethics committee. +The age of the subjects was ranged from 51 to 86 years old +(Male: 38, Female: 45). +Each scan contained 122-250 2D transverse US images with +resolution of 640*480. 7596 2D images from 40 scans were +manually delineated for MAB and LIB and labeled for healthy +or diseased (with plaque) by experienced sonographers for +further training of segmentation and classification network. All +Fig. 5. The illustration of the reprojection process. The centroid point was calculated by the segmented MAB mask for each slice in the volume. Then the line +segment crosses the centroid point was set to conduct the reprojection. The red and green line segment represent the different resample angle respectively. + + +(a) (b) + + +(c) (d) +Fig. 6. Ultrasound scan using the freehand imaging system. (a) a handheld +US scanner (left), a host laptop computer (middle) and an iPhone SE2 +(right). (b) Tracking system including a hub (left) and a RF/USB module +(right). (c) The sensor (left) and the magnetic source (right). (d) Ultrasound +scan using the freehand imaging system. + + +Get Centroid +0 +Theindexofthecolumn +n +GetCentroid +i-1 +Reprojection longitude image, reprojection angle=0o +i +i+1 +Get Centroid +Theindexofthecolumn +n +The index of the slice +6 +127 scans were labeled for healthy or diseased (with plaque) by +the same raters examining 2D images. In addition, stenosis +grade and plaque size of randomly selected 20 scans were +manually measured by expert using clinical 2D US device for +verification of the proposed system and algorithm. +B. Training Methods +25 scans (4694 2D images) were randomly chosen for CNN +training and 15 scans (2362 2D images) for validation in order +to build and verify the segmentation module. The original +images were resized to 224*224. All training process were +performed using Pytorch 1.5.1 and Python 3.7 on a NVIDIA +RTX 4000 GPU. The two networks were trained separately. For +the segmentation module, the applied data augmentation +strategies including gamma transformation, rotation, zoom, +horizontal and vertical flip, and Adam optimizer were used. The +network was trained for 100 epochs with learning rate and batch +size set to 0.005 and 8 respectively. For the diagnosis module, +the cropped and resized 2D US image segmented with the mask +and the corresponding vessel wall mask were used for network +training. Gamma transformation and horizontal & vertical flip +were applied for data augmentation. The diagnosis network was +trained for 50 epochs using Adam optimizer with learning rate +and batch size set to 0.005 and 64 respectively. +C. Diagnosis parameter measurement +To verify the regularized reconstruction and longitudinal +images reprojection algorithm, the longitudinal images from 20 +clinical patients with and without regularization were compared +with clinical images acquired by experienced sonographers +visually, and the projection angles were set as 𝜃 = +−30°, −15°, 0°, 15°, 30°. +The plaque length and thickness were manually measured on +the 3D pseudo volume, the reconstructed 3D volume and the +clinical images acquired by experienced sonographers +respectively, where 3D pseudo volume was the volume which +were stacked directly by the 2D US images sequence. The +manual measurement of plaque length and thickness was +conducted on the reprojected longitudinal images, among +which the reprojection angle was chosen based on the carotid +structural integrity and maximum stenosis grade. For plaque +size measurement in reprojected image of reconstructed 3D +volume, the pixel size was 0.2 × 0.2𝑚𝑚2. For the pseudo 3D +volume, the velocity of the swept was assumed constant, +therefore the pixel size of reprojected image was determined by +the distance of the swept which could be calculated by the +magnetic sensor. +The whole system in clinical metric measurement was also +verified by comparing stenosis rate automatically measured by +the +system +and +manually +measured +by +experienced +sonographers using clinical US device on 20 random clinical +atherosclerosis patients according to formula (9). +D. Evaluation Metrics and Statistic Analysis +The dice similarity coefficient (DSC) and 95% hausdorff +distance (HD95) were used to evaluate the performance of the +carotid sequence segmentation. DSC indicated the quantitative +metric of the overlap region between the ground truth and +prediction mask which was defined as follows: +𝐷𝑆𝐶 = 2(𝑃 ∩ 𝐿) +𝑃 ∪ 𝐿 +(10) +where P, L were the prediction mask and ground truth. The +hausdorff distance was defined as Eq (11), which indicated the +largest point-wise matching discrepancy: +𝐻𝐷(𝐴, 𝐵) = 𝑚𝑎𝑥(ℎ𝑑(𝐴, 𝐵), ℎ𝑑(𝐵, 𝐴)) +(11) +where +ℎ𝑑(𝐴, 𝐵) = 𝑚𝑎𝑥𝑎∈𝐴(𝑚𝑖𝑛𝑏∈𝐵||𝑎 − 𝑏||) +(12) +ℎ𝑑(𝐵, 𝐴) = 𝑚𝑎𝑥𝑏∈𝐵(𝑚𝑖𝑛𝑎∈𝐴||𝑏 − 𝑎||) +(13) + For the evaluation of the diagnosis module. The specificity, +sensitivity and accuracy were calculated for both 2D US image +and scans. + The mean absolute difference (MAD) and standard deviation +(SD) between results from the pseudo/reconstructed 3D +volumes and results from experienced sonographers were +investigated. The metrics in verification of the system, i.e., the +stenosis grade, were compared between manual or automatic +approach using the proposed +technique and manual +measurement using the clinical US device with the Pearson +correlation analysis. +IV. RESULTS +A. Segmentation and Diagnosis Accuracy +The comparison between nine typical segmented images +from U-Net and experienced sonographers was illustrated as +Fig. 7, and the images were selected from different scans at +some specific locations. Table I showed the average DSC and +HD95 between the ground truth and prediction results. +TABLE I. +THE RESULTS OF VESSEL SEGMENTATION +Metrics +category +MAB +Lumen +DSC +95.00% +93.30% +HD95(pixel) +4.34 +4.65 +Table II showed the contingency table of the validation set of +2362 2D transverse images, and the sensitivity, specificity and +accuracy were 0.73, 0.97 and 0.91 respectively. Table III +showed the diagnostic results of carotid atherosclerosis for all + + + + + + + + + + + +Fig. 7. Comparison of the auto segmentation from U-net (red) and manual +segmentation from ground truth (green). + + +7 +scans, and the sensitivity, specificity and accuracy of carotid +atherosclerosis detection was 0.71, 0.85 and 0.80 respectively. +TABLE II. +THE RESULTS OF DETECTION TEST FOR 2D IMAGES +Labels +Predictions +Positive (plaque) +Negative +Positive +(plaque) +454 +171 +Negative +50 +1687 +TABLE III. +THE DETECTION RESULTS OF CA FOR SCANS +Labels +Predictions +Positive (plaque) +Negative +Positive +(plaque) +25 +10 +Negative +10 +57 +B. Reconstruction and Reprojection Accuracy +Fig. 8 illustrated three representative examples of the +longitudinal images without regularization, with regularization +clinical US images acquired by experienced sonographers, and +the corresponding orientation information. The results revealed +that the regularized reconstructed volume was smoother with +less image artifacts. Fig. 9 demonstrated an example with large +fallback of trajectory, the results showed there were still +artifacts if directly apply the regularized algorithm and the +proposed re-rank algorithm could remove the reconstruction +artifacts. Fig. 10 illustrated the 3D volumes reconstructed from +the auto-segmentation and ground truth respectively. The +volumes were rendered by 3D-slicer (www.slicer.org). The +results showed that the segmentation module achieved good +agreement with human label. Furthermore, the sunken of the +lumen area indicated the existence of the plaque. Fig. 11 + + +Fig. 8. Illustration of the US longitudinal images and the corresponding orientation information from three carotid atherosclerosis patients (by rows). The +images in the first column were reconstructed without regularization algorithm while the images in the third column were reconstructed with regularization +algorithm. The second column demonstrated the smoother results of the proposed algorithm. The fourth column represents the images acquired by +sonographers using clinical US devices. The images in fifth column illustrate the corresponding original position information and the images in sixth column +show the regularized position information. + +Fig. 9. Illustration of the proposed re-rank algorithm, the first row +demonstrated the longitudinal image and corresponding position +information without regularized algorithm. The second row represented +the images which applied regularized algorithm and the third row showed +the images which used re-rank and regularized algorithm. + + +8 +demonstrated comparison among 5 projected images in +different angles ( 𝜃 = −30°, −15°, 0°, 15°, 30° ), the image +directly projected to sagittal plane and the manually acquired +image by expert from the same atherosclerosis patient. The +results showed that the projected images in different angles +could reveal more structures of the carotid than the images only +projected to sagittal plane. On the other hand, in Fig. 11, the +image in 15° projection angle was most consistent with the +clinical image obtained by expert using clinical US device, +which indicated that the reprojection of 3D volume could +simulate the different scan angles operated by expert to locate +the best observation view. +The plaque size (length and thickness) measured from the +pseudo volume, reconstructed volume and images acquired by +expert were compared in Table IV. The results showed good +agreement between the automatic measurement from the +reconstructed volume and the manual method, while the plaque +size measured by pseudo volume showed large difference with +the expert measurement. The results indicated that the 3D +reconstruction could reveal the true geometry and clinical +metric of the carotid artery. +TABLE IV. +MAD MEASUREMENTS (N=20) BETWEEN CLINICAL US DEVICE +AND THE PROPOSED TECHNIQUES + + + + + + + + + +Fig. 10. The 3D volumes from the auto-segmentation (the first row) and ground truth (the second row). The translucent outer wall represents the vessel wall +area, the inside red 3D volume represents the lumen area. The sunken of the lumen area indicated the existence of the plaque. The resolution of reconstruction +is set to 0.2×0.2×0.2 𝑚𝑚3. + + + + + + + + + +Fig. 11. Illustration of the projected +images in different angles, from left +top to right bottom were 5 projected +images (𝜃 = −30°, −15°, 0°, 15°, +30° ), direct sagittal image and +clinical image respectively. It could +be observed the sagittal image +missed the part of vessel wall (in the +red box) and the reprojected image +with 𝜃 = 15 ° showed the most +consistent structure of plague with +clinical image (in the green boxes +shows). + + +9 + +Plaque +length(mm) +Plaque +thickness(mm) +Plaque +length +(relative +error) +Plaque +thickness +(relative +error +3D +Reconstructed +volume +2.65±2.36 +0.842±0.617 +15.4%± +13.6% +26.0%± +13.2% +Direct stacked +Pesudo volume +6.54±7.23 +0.976±0.648 +40.0%± +48.0% +29.4%± +14.0% +C. Stenosis Measurement Accuracy +Fig. 12 demonstrated the linear correlation (r=0.762) of the +stenosis grade measured by the system and experienced +sonographers using the clinical US device on 20 carotid +atherosclerosis patients, which indicated the proposed +technique had the strong consistency with expert manual +approach in carotid atherosclerosis diagnosis. +V. DISCUSSION +In this study, we proposed a portable freehand 3D US +imaging technique for carotid artery diagnosis which could +achieve real 3D geometry of carotid artery, and the method +showed good agreements with manual measurement of stenosis +rate and classification of diseased and healthy case. The system +was transportable and less dependent on operator’s experience, +which make it possible for routine health check in different +environments such as community or rural area. In addition, the +3D reconstructed geometry could provide visualized carotid +artery structure for further atherosclerosis evaluation. +Since the large position variation or fallback movement +during scan would cause reconstruction artifacts, we designed +a standard scan protocol for 3D carotid US data acquisition and +analysis. The whole processing steps included automatic 3D US +data +acquisition, +MAB +and +LIB +segmentation, +3D +reconstruction, automatic classification and measurement. In +practice, the intermediate results of each step could be reviewed +and manually corrected by operator if necessary to ensure the +accurate final results. The diagnosis result was based on two +key points: one was the accurate segmentation of vessel area, +and the other was the correct reconstruction volume. The +segmentation determined the region of interest (ROI) used for +following analysis including automatic stenosis evaluation, +plaque size measurement and 3D geometry visualization. The +wrong mask might crop regions out of the carotid artery, +mislead the diagnosis network and cause confusing diagnosis +results. However, if the 3D volume was directly reconstructed +from +original +2D +frames +before +segmentation, +the +reconstruction artifacts around MAB and LIB such as +misplacement or severe blurring could lead to segmentation +error of vessels, especially for some cases with large position +variation as Fig. 13. showed. Therefore, we conducted +segmentation on the original 2D US image sequence before 3D +reconstruction to extract the vessel area firstly to reduce the +influence of reconstruction artifacts. +For the reconstruction process, the failure reconstruction +caused by large position variation could result in severe image +artifacts which totally deviated the structure of the carotid artery +as shown in Fig. 14 For the freehand US scan, theoretically, the +position information recorded by magnetic sensor would be +consistent with US probe motion i.e., the position of every US + +Fig. 12. correlation of stenosis grade between the manual measurement by +expert using the clinical US device and the automatic measurement from +the proposed technique on 20 carotid atherosclerosis patients. + + + + + +Fig. 14. Severe reconstruction artifacts caused by the large position +variation. The image in first row represented the reconstructed volume and +the orientation information with regularized algorithm while the images in +the second row represented the results without regularized algorithm. The +left image shows the transverse image of the locations in the reconstructed +volume marked in red boxes in the right image, the large distortion could +be observed in the image while the distortion was alleviated using the +regularized algorithm. + + +Fig. 13. Segmentation results on a transverse image collected from the 3D +volume reconstructed by the original image sequence (left) and on an +original transverse frame data (right). It could be observed that the severe +artifacts on the left image led to wrong segmentation result. + +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Stenosis grade measured by expert using clinicla US device +10 +image. However, the low precision of the sensor and inevitable +hand jitter would lead to the noticeable measurement +uncertainty of the position information along the scan direction +and influence the reconstruction accuracy. Therefore, we +adopted a novel total variation regularization algorithm to +smooth the track of the position information and decrease +distortion and disconnection of the image volume. The position +of the freehand scan can be regarded as continuous and +sequential array; therefore, the proposed regularization +algorithm could reduce the uncertainty by constructing and +minimize a regularized formulate in the manifold of Euclidean +transformations. Meanwhile, a re-rank strategy was designed to +solve the unordered image sequence caused by fallback +movement during scan. In the future, the reconstruction +accuracy could be further improved using the neural network. +After segmentation and reconstruction, the carotid artery +volume could be obtained for further analysis such as healthy +or diseased case diagnosis, plaque thickness, length area +measurement, +plaque +type +identification +and +stenosis +measurement etc. In the diagnosis module, the cropped and +resized images instead of the whole US images were used as the +input. Since the plaque was only located inside vessel wall area, +removing useless information outside the vessel wall could +accelerate network training and improve the detection accuracy. +On the other hand, there may be low intensity area in the vessel +region which could mislead the network and result in wrong +classification since negative sample (no plaque) usually had +low intensity in lumen area. Therefore, the MAB and LIB mask +were introduced to combine the morphological information +with original image information to improve the detection +accuracy. However, the proposed approach just utilized the +consecutive 2D reconstructed transverse US images to detect +plaque cases, thus some cases with small plaque size or severe +artifacts were wrongly classified as no plaque. In the future, we +will take the z axis information into account and use the whole +3D volume as input instead of detecting plaque by limited +consecutive transverse slices to improve the accuracy of the +diagnosis module. +We utilized a reprojection algorithm to project the carotid +artery volume to longitudinal planes, so that the clinical metric +such plaque length, thickness could be directly measured from +the 3D volume with no need of new acquisition in sagittal +direction. The traditional clinical carotid artery US examination +required appropriate positioning and angle between transducer +and neck, which greatly relies on the operator’s experience to +localize the plaque and obtain a high-quality US image, the +proposed reprojection approach in our method was not only +relatively convenient but could reveal the complete structure of +the carotid artery with only one scan, and the images obtained +by our automatic method achieved great agreement with the +images obtained by expert using clinical US device. +In segmentation module, we used U-Net to segment the +MAB and LIB in 2D US image sequence. Every image in the +sequence was treated as a single image for the segmentation +network. However, this approach didn’t exploit the context +information in the adjacent frames. In addition, some cases with +severe noise or shadowing would result in wrong segmentation +as Fig. 15 showed. In the future, 3D convolution will be +considered to correct the segmentation mistake by utilizing the +context information of the adjacent frames, and sample size will +be enlarged to improve the accuracy and robustness of the +segmentation algorithm. More 3D metrics such total plaque +volume, vessel wall volume, etc. would be evaluated for more +accurate validation. On the other hand, the learning-based 3D +reconstruction algorithm would be taken into account to +improve the performance of reconstruction. +VI. CONCLUSION +We have proposed an automatic 3D carotid artery imaging +and diagnosis technique specially designed for the portable +freehand ultrasound device. The technique applied a novel 3D +reconstructed algorithm and a robust segmentation algorithm +for automatic carotid atherosclerosis analysis. The results +demonstrated that the technique achieved good agreement with +manual expert examination on plaque diagnosis and stenosis +grade measurement, which showed the potential application on +fast carotid atherosclerosis examination and the follow-ups, +especially for those scenarios where professional medical +device and experienced clinicians are hard to acquire such as +rural area or community with large population. +ACKNOWLEDGEMENT + This work was sponsored by Natural Science Foundation of +China (NSFC) under Grant No.12074258. +REFERENCES +[1] +M. L. Flaherty et al., “Carotid artery stenosis as a cause of stroke,” +Neuroepidemiology, vol. 40, no. 1, pp. 36–41, 2013. +[2] +L.-Y. Ma et al., “China cardiovascular diseases report 2018: an +updated summary,” J. Geriatr. Cardiol. JGC, vol. 17, no. 1, p. 1, +2020. +[3] +H. R. Underhill, T. S. Hatsukami, Z. A. Fayad, V. Fuster, and C. +Yuan, “MRI of carotid atherosclerosis: clinical implications and future +directions,” Nat. Rev. Cardiol., vol. 7, no. 3, pp. 165–173, 2010. +[4] +M. Wintermark et al., “High-resolution CT imaging of carotid artery +atherosclerotic plaques,” Am. J. Neuroradiol., vol. 29, no. 5, pp. 875– +882, 2008. +[5] +P. J. Nederkoorn, Y. van der Graaf, and M. M. Hunink, “Duplex +ultrasound and magnetic resonance angiography compared with +digital subtraction angiography in carotid artery stenosis: a systematic +review,” Stroke, vol. 34, no. 5, pp. 1324–1331, 2003. +[6] +Y. Inaba, J. A. Chen, and S. R. Bergmann, “Carotid plaque, compared +with carotid intima-media thickness, more accurately predicts +coronary artery disease events: a meta-analysis,” Atherosclerosis, vol. +220, no. 1, pp. 128–133, 2012. +[7] +M. W. Lorenz, H. S. Markus, M. L. Bots, M. Rosvall, and M. Sitzer, +“Prediction of clinical cardiovascular events with carotid intima- + + +Fig. 15. Two representative wrong segmentation examples. The red line +represented the automatic segmentation results by the segmentation +module and the green line represented the human labels. The plaque was +identified as the adventitia in the first case. In the second case, the vessel +wall structure was disappeared and the segmentation network resulted in +wrong segmentation. + + +11 +media thickness: a systematic review and meta-analysis,” Circulation, +vol. 115, no. 4, pp. 459–467, 2007. +[8] +A. Fenster, G. Parraga, and J. Bax, “Three-dimensional ultrasound +scanning,” Interface Focus, vol. 1, no. 4, pp. 503–519, 2011. +[9] +T. Wannarong et al., “Progression of carotid plaque volume predicts +cardiovascular events,” Stroke, vol. 44, no. 7, pp. 1859–1865, 2013. +[10] +A. Fenster, C. Blake, I. Gyacskov, A. Landry, and J. Spence, “3D +ultrasound analysis of carotid plaque volume and surface +morphology,” Ultrasonics, vol. 44, pp. e153–e157, 2006. +[11] +G. C. Makris, A. Lavida, M. Griffin, G. Geroulakos, and A. N. +Nicolaides, “Three-dimensional ultrasound imaging for the evaluation +of carotid atherosclerosis,” Atherosclerosis, vol. 219, no. 2, pp. 377– +383, 2011. +[12] +K. AlMuhanna et al., “Carotid plaque morphometric assessment with +three-dimensional ultrasound imaging,” J. Vasc. Surg., vol. 61, no. 3, +pp. 690–697, 2015. +[13] +R. M. Botnar, M. Stuber, K. V. Kissinger, W. Y. Kim, E. Spuentrup, +and W. J. Manning, “Noninvasive coronary vessel wall and plaque +imaging with magnetic resonance imaging,” Circulation, vol. 102, no. +21, pp. 2582–2587, 2000. +[14] +M. Herder, S. H. Johnsen, K. A. Arntzen, and E. B. Mathiesen, “Risk +factors for progression of carotid intima-media thickness and total +plaque area: a 13-year follow-up study: the Tromsø Study,” Stroke, +vol. 43, no. 7, pp. 1818–1823, 2012. +[15] +B. Chiu, M. Egger, J. D. Spence, G. Parraga, and A. Fenster, +“Quantification of carotid vessel wall and plaque thickness change +using 3D ultrasound images,” Med. Phys., vol. 35, no. 8, pp. 3691– +3710, 2008. +[16] +X. Yang, J. Jin, W. He, M. Yuchi, and M. Ding, “Segmentation of the +common carotid artery with active shape models from 3D ultrasound +images,” in Medical Imaging 2012: Computer-Aided Diagnosis, 2012, +vol. 8315, p. 83152H. +[17] +A. M. A. Lorza et al., “Carotid artery lumen segmentation in 3D free- +hand ultrasound images using surface graph cuts,” in International +conference on medical image computing and computer-assisted +intervention, 2013, pp. 542–549. +[18] +J. de Ruijter, M. van Sambeek, F. van de Vosse, and R. Lopata, +“Automated 3D geometry segmentation of the healthy and diseased +carotid artery in free-hand, probe tracked ultrasound images,” Med. +Phys., vol. 47, no. 3, pp. 1034–1047, 2020. +[19] +M. Jiang, J. D. Spence, and B. Chiu, “Segmentation of 3D ultrasound +carotid vessel wall using U-Net and segmentation average network,” +in 2020 42nd Annual International Conference of the IEEE +Engineering in Medicine & Biology Society (EMBC), 2020, pp. 2043– +2046. +[20] +R. Zhou, A. Fenster, Y. Xia, J. D. Spence, and M. Ding, “Deep +learning-based carotid media-adventitia and lumen-intima boundary +segmentation from three-dimensional ultrasound images,” Med. Phys., +vol. 46, no. 7, pp. 3180–3193, 2019. +[21] +J. De Ruijter, J. J. Muijsers, F. N. Van de Vosse, M. R. Van Sambeek, +and R. G. Lopata, “A generalized approach for automatic 3-D +geometry assessment of blood vessels in transverse ultrasound images +using convolutional neural networks,” IEEE Trans. Ultrason. +Ferroelectr. Freq. Control, vol. 68, no. 11, pp. 3326–3335, 2021. +[22] +L. van Knippenberg, R. J. van Sloun, M. Mischi, J. de Ruijter, R. +Lopata, and R. A. Bouwman, “Unsupervised domain adaptation +method for segmenting cross-sectional CCA images,” Comput. +Methods Programs Biomed., vol. 225, p. 107037, 2022. +[23] +C. Azzopardi, K. P. Camilleri, and Y. A. Hicks, “Bimodal automated +carotid ultrasound segmentation using geometrically constrained deep +neural networks,” IEEE J. Biomed. Health Inform., vol. 24, no. 4, pp. +1004–1015, 2020. +[24] +R. Zhou et al., “A voxel-based fully convolution network and +continuous max-flow for carotid vessel-wall-volume segmentation +from 3D ultrasound images,” IEEE Trans. Med. Imaging, vol. 39, no. +9, pp. 2844–2855, 2020. +[25] +R. Zhou et al., “Deep learning-based measurement of total plaque area +in B-mode ultrasound images,” IEEE J. Biomed. Health Inform., vol. +25, no. 8, pp. 2967–2977, 2021. +[26] +R. Zhou et al., “Deep learning-based carotid plaque segmentation +from B-mode ultrasound images,” Ultrasound Med. Biol., vol. 47, no. +9, pp. 2723–2733, 2021. +[27] +Y. Xia, X. Cheng, A. Fenster, and M. Ding, “Automatic classification +of carotid ultrasound images based on convolutional neural network,” +in Medical Imaging 2020: Computer-Aided Diagnosis, 2020, vol. +11314, p. 1131441. +[28] +W. Ma et al., “Multilevel strip pooling-based convolutional neural +network for the classification of carotid plaque echogenicity,” +Comput. Math. Methods Med., vol. 2021, 2021. +[29] +H. Shen, W. Zhang, H. Wang, G. Ding, and J. Xie, “NDDR-LCS: A +Multi-Task Learning Method for Classification of Carotid Plaques,” in +2020 IEEE International Conference on Image Processing (ICIP), +2020, pp. 2461–2465. +[30] +Y. Zhao, J. D. Spence, and B. Chiu, “Three-dimensional ultrasound +assessment of effects of therapies on carotid atherosclerosis using +vessel wall thickness maps,” Ultrasound Med. Biol., vol. 47, no. 9, pp. +2502–2513, 2021. +[31] +R. Zhou, W. Ma, A. Fenster, and M. Ding, “U-Net based automatic +carotid plaque segmentation from 3D ultrasound images,” in Medical +Imaging 2019: Computer-Aided Diagnosis, 2019, vol. 10950, pp. +1119–1125. +[32] +L. Saba et al., “Ultrasound-based carotid stenosis measurement and +risk stratification in diabetic cohort: a deep learning paradigm,” +Cardiovasc. Diagn. Ther., vol. 9, no. 5, p. 439, 2019. +[33] +M. Biswas et al., “Two-stage artificial intelligence model for jointly +measurement of atherosclerotic wall thickness and plaque burden in +carotid ultrasound: A screening tool for cardiovascular/stroke risk +assessment,” Comput. Biol. Med., vol. 123, p. 103847, 2020. +[34] +T. Wen et al., “An accurate and effective FMM-based approach for +freehand 3D ultrasound reconstruction,” Biomed. Signal Process. +Control, vol. 8, no. 6, pp. 645–656, 2013. +[35] +O. V. Solberg, F. Lindseth, H. Torp, R. E. Blake, and T. A. N. Hernes, +“Freehand 3D ultrasound reconstruction algorithms—a review,” +Ultrasound Med. Biol., vol. 33, no. 7, pp. 991–1009, 2007. +[36] +H. R. Roth et al., “A new 2.5 D representation for lymph node +detection using random sets of deep convolutional neural network +observations,” in International conference on medical image +computing and computer-assisted intervention, 2014, pp. 520–527. +[37] +D.-H. Lee and others, “Pseudo-label: The simple and efficient semi- +supervised learning method for deep neural networks,” in Workshop +on challenges in representation learning, ICML, 2013, vol. 3, no. 2, p. +896. +[38] +O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional +networks for biomedical image segmentation,” in International +Conference on Medical image computing and computer-assisted +intervention, 2015, pp. 234–241. +[39] +H.-B. Chen, R. Zheng, L.-Y. Qian, F.-Y. Liu, S. Song, and H.-Y. +Zeng, “Improvement of 3-D Ultrasound Spine Imaging Technique +Using Fast Reconstruction Algorithm,” IEEE Trans. Ultrason. +Ferroelectr. Freq. Control, vol. 68, no. 10, pp. 3104–3113, 2021. +[40] +S. Song, Y. Huang, J. Li, M. Chen, and R. Zheng, “Development of +Implicit Representation Method for Freehand 3D Ultrasound Image +Reconstruction of Carotid Vessel,” in 2022 IEEE International +Ultrasonics Symposium (IUS), 2022, pp. 1–4. +[41] +M. Esposito et al., “Total variation regularization of pose signals with +an application to 3D freehand ultrasound,” IEEE Trans. Med. +Imaging, vol. 38, no. 10, pp. 2245–2258, 2019. +[42] +J. Wu et al., “Deep morphology aided diagnosis network for +segmentation of carotid artery vessel wall and diagnosis of carotid +atherosclerosis on black-blood vessel wall MRI,” Med. Phys., vol. 46, +no. 12, pp. 5544–5561, 2019. +[43] +H. Chen, R. Zheng, E. Lou, and L. H. Le, “Compact and Wireless +Freehand 3D Ultrasound Real-time Spine Imaging System: A pilot +study.,” in Annual International Conference of the IEEE Engineering +in Medicine and Biology Society. IEEE Engineering in Medicine and +Biology Society. Annual International Conference, 2020, vol. 2020, +pp. 2105–2108. + + diff --git a/7tE1T4oBgHgl3EQfTwM_/content/tmp_files/load_file.txt b/7tE1T4oBgHgl3EQfTwM_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d708edea325008caa033ede314f69ef2111d09b --- /dev/null +++ b/7tE1T4oBgHgl3EQfTwM_/content/tmp_files/load_file.txt @@ -0,0 +1,877 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf,len=876 +page_content='1 Abstract— Objective: The objective of this study is to develop a deep-learning based detection and diagnosis technique for carotid atherosclerosis using a portable freehand 3D ultrasound (US) imaging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Methods: A total of 127 3D carotid artery datasets were acquired using a portable 3D US imaging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' A U-Net segmentation network was firstly applied to extract the carotid artery on 2D transverse frame, then a novel 3D reconstruction algorithm using fast dot projection (FDP) method with position regularization was proposed to reconstruct the carotid artery volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Furthermore, a convolutional neural network was used to classify the healthy case and diseased case qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3D volume analysis including longitudinal reprojection algorithm and stenosis grade measurement algorithm was developed to obtain the clinical metrics quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Results: The proposed system achieved sensitivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='714, specificity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='851 and accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='803 respectively in diagnosis of carotid atherosclerosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The automatically measured stenosis grade illustrated good correlation (r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='762) with the experienced expert measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Conclusion: the developed technique based on 3D US imaging can be applied to the automatic diagnosis of carotid atherosclerosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Significance: The proposed deep-learning based technique was specially designed for a portable 3D freehand US system, which can provide carotid atherosclerosis examination more conveniently and decrease the dependence on clinician’s experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Index Terms—3D ultrasound imaging, automatic carotid atherosclerosis diagnosis, carotid artery segmentation, reconstruction with regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' INTRODUCTION AROTID atherosclerosis is one of the major causes of stroke which is the world’s second leading cause of death [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The prevalence rate of carotid atherosclerosis is 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='2% in Chinese people over 40 years old [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The pathological features of carotid atherosclerosis are increase of intima-media thickness and appearance of atherosclerosis plaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Magnetic resonance imaging (MRI), computed tomography angiography This work was sponsored by Natural Science Foundation of China (NSFC) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='12074258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (Jiawen Li and Yunqian Huang are co-first authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=') (Corresponding authors: Rui Zheng, Man Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=') Jiawen Li, Sheng Song, Duo Xu and Haibin Zhang are with School of Information Science and Technology, ShanghaiTech University, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Hongbo Chen is with School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China, also with Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 200050, (CTA) and digital subtraction angiography (DSA) are several commonly used methods for visualizing and characterizing carotid artery features [3]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' However, these methods still have some limitations during application due to invasiveness, ionizing radiation, heavy equipment etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' and the approaches are very time-consuming and expensive which can’t satisfy the need of large scale of examinations in different environments especially for community and countryside areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2D Ultrasound (US), as a non-invasive and low-cost method, is widely used in the examination of carotid plaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' However, there are several disadvantages of traditional 2D US in the current ultrasound examination of carotid atherosclerosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (1) It is mainly carried out by experienced sonographers in hospital, and becomes a huge burden for health care system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (2) Routine health check is difficult for carotid atherosclerosis patients especially in rural or undeveloped area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (3) Routine ultrasound examination is a tedious, laborious, experience-dependent work for sonographers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (4) Clinically, some metrics such as intima- media thickness (IMT), plaque thickness, plaque area, usually assess the severity of the carotid atherosclerosis in 2D US images, which is prone to variability and lack of 3D morphology of carotid plaque [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3D US carotid artery imaging approaches mainly include mechanical scanning and tracked freehand scanning using various sensors e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', magnetic tracked senor, optical tracked sensor, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', [8] which can provide plaque volume estimation, 3D morphology of plaque and other 3D metrics for carotid atherosclerosis diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The 3D techniques are found to be more accurate to evaluate the progress of carotid atherosclerosis [9]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Therefore, it is of great importance to develop a portable, reliable and cost- effective automatic ultrasound diagnostic technique for carotid atherosclerosis screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The automatic diagnosis of carotid atherosclerosis focuses on finding the biomarkers on the ultrasound images, for example China, and also with University of Chinese Academy of Sciences, Beijing 100049, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Yunqian Huang and Junni Shi are with Tongren Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Man Chen is with Tongren Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai, China (e-mail: maggiech1221@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='com) Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Rui Zheng is with School of Information Science and Technology, Shanghai Engineering Research Center of Energy Efficient and Custom AI IC, ShanghaiTech University, Shanghai, China (phone: 86 21-2068 4452, e-mail: zhengrui@shanghaitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='cn) Automatic Diagnosis of Carotid Atherosclerosis Using a Portable Freehand 3D Ultrasound Imaging System Jiawen Li, Yunqian Huang, Sheng Song, Hongbo Chen, Junni Shi, Duo Xu, Haibin Zhang, Man Chen*, Rui Zheng* C 2 vessel wall area, vessel wall volume or total plaque volume [13]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' These biomarkers are all bounded by the two boundaries of vessels, the media-adventitia boundary (MAB) and the lumen-intima boundary (LIB), thus identifying these two boundaries is an important issue during the carotid atherosclerosis diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In recent years, deep learning methods has achieved excellent performance in medical image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [16]–[18]designed a novel adaptive triple loss for carotid artery segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' To utilize 3D information in 3D volume of carotid artery, Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [19] introduced a fusion module to the U-Net segmentation network and yielded promising performance on carotid segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [20] proposed a deep learning-based MAB and LIB segmentation method, and a dynamic convolutional neural network (CNN) were applied to image patches in every slice of the 3D US images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' LIB segmentation was performed by U-Net based on the masks of the MAB since the LIB is inside the MAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The method achieved high accuracy but initial anchor points were still manually placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ruijter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [21] created a generalized method to segment LIB using CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Several U- Nets were compared and the experiments showed that the combination of various vessels such as radial, ulnar artery, or cephalic vein improved the segmentation performance of carotid artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' After segmentation, a 3D-geometry can be obtained for further therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Van Knippenberg et al [22] proposed an unsupervised learning method to solve the lack of data in carotid segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Azzopardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [23] designed a novel geometrically constrained loss functions and received improved segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [24] proposed a voxel based 3D segmentation neural network to segment the MAB and LIB in 3D volume directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Although the proposed algorithm achieved high accuracy with fast process, user’s interaction is yet required to identify ROI in the first and last slice of the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' After region of interest (ROI) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', carotid artery is identified, further analysis needs to be performed to get significant clinical information for carotid atherosclerosis diagnosis such as the existence of plaque, carotid stenosis grade, type of the plaque, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [25],[26] applied 8 different backbone and UNet++ segmentation algorithm trained on 2D longitudinal US images to segment the plaque region and calculate the total plaque area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [27] employed a CNN to categorize segmented carotid images into normal cases, thickening vessel wall cases and plaque cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [28] proposed a multilevel strip pooling-based convolutional neural network to investigate the echogenicity of plaque which was found to be closely correlated with the risk of stroke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [29] proposed a multi task learning method, the authors combined ultrasound reports and plaque type label to train a CNN to classify four different plaque type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [30] utilized a novel vessel wall thickness mapping algorithm to evaluate the therapeutical performance on carotid atherosclerosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [31] utilized the unsupervised pretrained parameters of U-Net to train a plaque segmentation network with a small 3D carotid artery ultrasound dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Saba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [32] used a deep learning based method to measure the carotid stenosis, three deep learning based systems were evaluated on 407 US dataset, and achieved AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='90, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='94 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='86 on the longitudinal US images respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Biswas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [33] proposed a two-stage artificial intelligence model for jointly measurement of atherosclerotic wall thickness and plaque burden in longitudinal US images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The results showed that the proposed method achieved the lowest error compared to previous method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The current 3D carotid imaging device was mainly based on mechanical system and hard to transport which was almost impossible to apply in community or rural area, therefore the portable freehand 3D ultrasound imaging system was required which can be easily applied for various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' However, for the freehand 3D ultrasound reconstruction, the requested small voxel size and various noise would lead to reconstruction artifacts[34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' On the other hand, the clinicians in different scenarios were usually inexperienced so that the diagnosis results might be inaccurate and hard to reproduce compared with sonographers in clinical ultrasound department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In this paper, we developed a new detection and classification technique based on deep-learning algorithms for carotid atherosclerosis diagnosis which can be employed to a portable freehand 3D US imaging system for fast screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Compared to other 3D ultrasound carotid artery imaging methods mainly focusing on carotid vessel wall segmentation [18], [20], [21], [24], the proposed method aimed at exploring an automatic and experience-independent technique and framework for fast carotid arteriosclerosis diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The main contributions are outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Firstly, a portable freehand 3D US carotid imaging and diagnosis framework including deep-learning based segmentation, 3D reconstruction and automatic volume analysis was developed for fast carotid atherosclerosis diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Secondly, a novel position regularization algorithm was designed to reduce the reconstruction error caused by freehand scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Lastly, post analysis including automatic reprojection and stenosis measurement from 3D volume data provided visible qualitative results and quantitative results for atherosclerosis diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' METHODS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1 showed the overview of data processing procedure including transverse image segmentation, 3D volume reconstruction, detection of carotid atherosclerosis and 3D carotid volume analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' MAB and LIB Segmentation Three consecutive frames were concatenated in channel dimension which is proved to be useful to improve the segmentation accuracy [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Since the adjacent frames contained lots of redundant information, the pseudo labels were generated using pseudo- labeling method to reduce the work load [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' One of every 5 neighbor frames were selected to be manually labeled by experienced sonographers and the other four frames were inferred by the network which was trained using the labeled frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' All generated pseudo labels were checked visually, the labels would be corrected if the segmentation is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The intensity of the image was normalized to [0,1] as follows: 𝐼 = 𝐼 − 𝐼𝑚𝑖𝑛 𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛 (1) 3 where I represented the intensity of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Imax and Imin represent the max and minimum value of the intensity in the US image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' All images and corresponding labels were resized to 224*224 for segmentation network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' U-Net was employed to segment the MAB and LIB in the transverse US image sequence [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The architecture of the network was illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The segmentation module consisted of two symmetrical sub-module which were encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The number of channels for each convolutional layer were set to (64, 128, 256, 512, 512).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Each convolutional layer was followed by a batch normalization module and a rectification linear unit (ReLU) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The two modules were connected using skip connection to exploit all resolution features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The loss function of the segmentation module was the combination of DSC loss and cross-entropy loss: 𝐿𝑜𝑠𝑠 = 𝐿𝑜𝑠𝑠𝑑𝑖𝑐𝑒 + 𝐿𝑜𝑠𝑠𝑐𝑒 (2) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3D Reconstruction with Regularization After the MAB and LIB were identified in every slice of US image sequence, the 3D carotid artery volume was reconstructed using the Fast Dot Projection (FDP) method [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' However, some disturbances caused by the low precision of the magnetic sensor, inevitable hand shaking and breathing movement during carotid swept, would lead to the reconstruction errors and artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The major problem was the repeated acquisition at the same or very close positions, and it caused large uncertainty at volume voxels and discontinuity in the reconstructed volume [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' To improve the image quality and decrease the uncertainty of 3D reconstructed volume, a total variation regularization [41] method was integrated with FDP reconstruction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (1) For all the position information obtained from 3DUS device, it could be formulated as a set of rotation matrix 𝑅 and a translation 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The tuple (𝑹, 𝒕) consisting of all 𝑅 and 𝑡 formed the special Euclidean group 𝑆𝐸(3) which was the semi-direct product of the rotation group 𝑆𝑂(3) and the translation group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Therefore, the 𝑆𝐸(3) can be formulated as: 𝑆𝐸(3) = {(𝑅 𝑡 0 1) : 𝑅 ∈ 𝑆𝑂(3), 𝑡 ∈ ℝ3} (3) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The architecture of the segmentation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The pipeline of the proposed system and corresponding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The top row demonstrated the process of the data acquisition, extraction of ROI and 3D reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The bottom row represented the process of 3D carotid volume analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The original image sequence and corresponding position information were firstly obtained by the 3D US device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' U-Net segmentation algorithm and regularized Fast-Dot Projection algorithm was applied to extract the ROI and 3D carotid volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Then 3D carotid volume analysis included automatic stenosis grade measurement, longitudinal image reprojection and healthy/diseased cases classification was conducted based on the reconstructed volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=':Conv+Batchnorm+ReLU × 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=':Maxpooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=': Upsampling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=':Concatenate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=':1×1 ConvSegmented masks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Data collection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Pre-processed images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='of LIB and MAB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Segmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Fast-Dot Projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Reconstruction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Reconstruction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Position ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Variation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Fast-Dot Projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Regularization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Reconstruction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Reconstruction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Stenosis Grade ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Stenosis rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='30° ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Measurement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Reprojection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='0° ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Exist Plaque ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Diagnosis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='or Not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='+30° ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='Projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='(2) The position signal obtained by the 3DUS system could ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='be considered as a set of entries which forms a vector 𝑷 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='(𝒑𝟏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 𝒑𝟐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' … ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 𝒑𝒌) ∈ 𝑴𝒌,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' where 𝑘 was the number of entries and 𝑘 ∈ 𝑁,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 𝑀𝑘 was a manifold and 𝑀 = 𝑆𝐸(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Another signal set X were considered to be found when the following formula is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' E(𝐱) = D(𝐱, 𝐩) + αR(𝐱), α > 0 (4) Where 𝐷(𝑥, 𝑝) was the penalizing term to reduce the variation between original signal P and resulted signal X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 𝑅(𝒙) was a regularized term to penalize the position saltation in the signal X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (3) The deviation penalized term D(x,p) could be defined as: 𝐷(𝐱, 𝐟) = ∑ 𝑘 𝑖=1 (ℎ ∘ 𝑑)(𝐱𝑖, 𝐩𝑖) (5) Where d(xi,pi) was the length of the geodesic which was defined as a shortest path on 𝑀 between two pose p and q [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' ℎ was defined as following: ℎ(𝑠) = {𝑠2, 𝑠 < 1/√2 √2𝑠 − 1/2, otherwise (6) Which was the Huber-Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (4) For the regularized term, it could be defined as the following: 𝑅(𝐱) = ∑ 𝑘−1 𝑖=1 (ℎ ∘ 𝑑)(𝐱𝑖, 𝐱𝑖+1) (7) Where d(xi,xi+1) could be considered as the first-order forward difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The optimize problem in (4) could be solved using a cyclic proximal point algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' However, the original regularized algorithm couldn’t handle the scanning positions with large backward movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In this case, the position array was not sequential according to the coordinates, therefore pose re-rank algorithm was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Concretely, considering the centroid point of every frame from the 2D segmented image sequence as 𝑪𝒌 = (𝒄𝟏, 𝒄𝟐, … , 𝒄𝒌) , the PCA (principal components analysis) algorithm was conducted in 𝐶𝑘 and a new matrix 𝐷𝑘 was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The first column of the matrix was the principal vector 𝑣𝑘, then a set of vectors 𝑐𝑑 could be acquired by projecting every centroid vector 𝑐𝑘 to 𝑣𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 𝒄𝒅 = 𝒄𝑘 − 𝒄𝒌 ⋅ 𝑣𝑘 𝑣𝑘 ⋅ 𝑣𝑘 𝑣𝑘 (8) The new position sequence was finally obtained by sorting the l2-norm of the 𝒄𝒅 set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Carotid Atherosclerosis Diagnosis The US scans including the segmented and reconstructed volume were classified into healthy case and carotid atherosclerosis case using a diagnosis network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3, there were two inputs for the diagnosis module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' It had been proved that the morphological information was helpful for the network to classify the normal or abnormal (diseased) image [42], therefore the mask of LIB and MAB extracted from each slice of the reconstructed volume was used as one input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The other input was the cropped ROI which was determined by the max bounding rectangular of the mask, and in the cropped image, the intensity in the region between LIB and MAB was set to the original value, while the region inside lumen and outside vessel wall were set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' For each input stream, it consisted of three repeated blocks, each block consisted of two consequent basic convolutional sub-block and a max pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The basic convolutional sub-block was composed of a convolutional layer, a batch normalization module and a linear rectification unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The number of channels for each repeated block were set to (24, 48, 96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The fusion block concatenated the high-level features of two streams and combined information by introducing a basic convolutional sub-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' After fusion block, the remaining layers were global average pooling (GAP) layers and a fully connected layer to output the diagnosis result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' We used focal loss in the diagnosis module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The scan would be diagnosed as a carotid atherosclerosis case if the consecutive 5 transverse slices from the reconstructed volume were classified as existing plaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3D Carotid Volume Analysis The clinical diagnostic parameters such as plaque thickness, plaque length, stenosis grade, plaque area, plaque type, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' can be directly calculated from the reconstructed carotid artery volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' To validate accuracy of the proposed method, the longitudinal US images of carotid artery were obtained by projecting the volume in different angles, and the stenosis grade was calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Stenosis rate was usually used to evaluate the stenosis grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' For the slices which were diagnosed as atherosclerosis, stenosis degree can be evaluated using the LIB and MAB masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The diameter stenosis rate was usually calculated to evaluate the stenosis grade in clinic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' We denote it as 𝑆𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 = 𝐿𝑤𝑎𝑙𝑙 𝐿𝑤𝑎𝑙𝑙 + 𝐿𝑙𝑢𝑚𝑒𝑛 (9) where 𝐿 represented the length of respective area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The metric was ranged from 0 to 1, the greater number indicated the more severe stenosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The length of vessel wall 𝐿𝑤𝑎𝑙𝑙 and length of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The architecture of the diagnosis module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 4 The illustration of the approach to calculate the diameter stenosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 24X128X128 128X128 48X64X64 96X32X32 96X16X16 96X16X16 Exist plaque 24X128X128 48X64X64 or not 96X32X32 :Conv+Batchnorm+ReLU 96X16X16 :Maxpooling 128X128 :Global average pooling :Concatenate + :Fully connectRadially Sampling :lumen :Vessel Wall 5 lumen 𝐿𝑙𝑢𝑚𝑒𝑛 were illustrated as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The diameter stenosis rate was the max 𝑆𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 which was calculated using all points in MAB boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The longitudinal carotid US images were usually used to calculate plaque size and evaluate the morphology of plaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Since the carotid artery is curved volume, the direct projection along a fixed axis may lead to missing of some structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Therefore, centroid points of carotid artery in transverse slices were selected to determine the projection plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Specifically, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 5, denoting the centroid point of i-th slice in the volume as 𝐶𝑖, the line which was 𝜃 degree angled with y- axis through the centroid point 𝐶𝑖 , was sampled as the i-th column of projected longitude image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In our experiment, the longitudinal US images were obtained by reprojecting the 3D carotid volume at the angles of 0°, ±15° and ±30°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' EXPERIMENTAL SETUP A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Data Acquisition and 3D Ultrasound Scan A portable freehand 3D ultrasound imaging system was used to obtain three-dimensional images of carotid artery as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The system consisted of a 2D linear probe (Clarius, L738-K, Canada), an electromagnetic tracking system (Polhemus, G4 unit, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='A) and a host laptop computer (Intel i7-8700k CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='70GHz, 32GB RAM) [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The 2D transverse US images were acquired by the probe while the corresponding position and orientation information were captured by the magnetic sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The images and orientation were acquired with a frame rate of 24 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' During the acquisition, the subjects took the supine position and was scanned as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 6 (d), and the probe swept consistently along the long axis of common carotid artery from the proximal end to the distal end at the speed of approximate 10-15 seconds per scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' To reduce the reconstruction artifacts, fallback along the swept direction and large movement normal to the swept direction should be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The inclusion criteria were based on visible plaques which was identified by expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The stenosis grade larger than 70% was excluded to the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' A total of 127 3D carotid artery scans from 83 subjects with stenosis range from 0% to 70% were obtained from local hospital, and all subjects consented to participate in this experiment, which was approved by the local ethics committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The age of the subjects was ranged from 51 to 86 years old (Male: 38, Female: 45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Each scan contained 122-250 2D transverse US images with resolution of 640*480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 7596 2D images from 40 scans were manually delineated for MAB and LIB and labeled for healthy or diseased (with plaque) by experienced sonographers for further training of segmentation and classification network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' All Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The illustration of the reprojection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The centroid point was calculated by the segmented MAB mask for each slice in the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Then the line segment crosses the centroid point was set to conduct the reprojection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The red and green line segment represent the different resample angle respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ultrasound scan using the freehand imaging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (a) a handheld US scanner (left), a host laptop computer (middle) and an iPhone SE2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (b) Tracking system including a hub (left) and a RF/USB module (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (c) The sensor (left) and the magnetic source (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' (d) Ultrasound scan using the freehand imaging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Get Centroid 0 Theindexofthecolumn n GetCentroid i-1 Reprojection longitude image, reprojection angle=0o i i+1 Get Centroid Theindexofthecolumn n The index of the slice 6 127 scans were labeled for healthy or diseased (with plaque) by the same raters examining 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In addition, stenosis grade and plaque size of randomly selected 20 scans were manually measured by expert using clinical 2D US device for verification of the proposed system and algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Training Methods 25 scans (4694 2D images) were randomly chosen for CNN training and 15 scans (2362 2D images) for validation in order to build and verify the segmentation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The original images were resized to 224*224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' All training process were performed using Pytorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='1 and Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='7 on a NVIDIA RTX 4000 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The two networks were trained separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' For the segmentation module, the applied data augmentation strategies including gamma transformation, rotation, zoom, horizontal and vertical flip, and Adam optimizer were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The network was trained for 100 epochs with learning rate and batch size set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='005 and 8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' For the diagnosis module, the cropped and resized 2D US image segmented with the mask and the corresponding vessel wall mask were used for network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Gamma transformation and horizontal & vertical flip were applied for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The diagnosis network was trained for 50 epochs using Adam optimizer with learning rate and batch size set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='005 and 64 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Diagnosis parameter measurement To verify the regularized reconstruction and longitudinal images reprojection algorithm, the longitudinal images from 20 clinical patients with and without regularization were compared with clinical images acquired by experienced sonographers visually, and the projection angles were set as 𝜃 = −30°, −15°, 0°, 15°, 30°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The plaque length and thickness were manually measured on the 3D pseudo volume, the reconstructed 3D volume and the clinical images acquired by experienced sonographers respectively, where 3D pseudo volume was the volume which were stacked directly by the 2D US images sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The manual measurement of plaque length and thickness was conducted on the reprojected longitudinal images, among which the reprojection angle was chosen based on the carotid structural integrity and maximum stenosis grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' For plaque size measurement in reprojected image of reconstructed 3D volume, the pixel size was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='2𝑚𝑚2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' For the pseudo 3D volume, the velocity of the swept was assumed constant, therefore the pixel size of reprojected image was determined by the distance of the swept which could be calculated by the magnetic sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The whole system in clinical metric measurement was also verified by comparing stenosis rate automatically measured by the system and manually measured by experienced sonographers using clinical US device on 20 random clinical atherosclerosis patients according to formula (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Evaluation Metrics and Statistic Analysis The dice similarity coefficient (DSC) and 95% hausdorff distance (HD95) were used to evaluate the performance of the carotid sequence segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' DSC indicated the quantitative metric of the overlap region between the ground truth and prediction mask which was defined as follows: 𝐷𝑆𝐶 = 2(𝑃 ∩ 𝐿) 𝑃 ∪ 𝐿 (10) where P, L were the prediction mask and ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The hausdorff distance was defined as Eq (11), which indicated the largest point-wise matching discrepancy: 𝐻𝐷(𝐴, 𝐵) = 𝑚𝑎𝑥(ℎ𝑑(𝐴, 𝐵), ℎ𝑑(𝐵, 𝐴)) (11) where ℎ𝑑(𝐴, 𝐵) = 𝑚𝑎𝑥𝑎∈𝐴(𝑚𝑖𝑛𝑏∈𝐵||𝑎 − 𝑏||) (12) ℎ𝑑(𝐵, 𝐴) = 𝑚𝑎𝑥𝑏∈𝐵(𝑚𝑖𝑛𝑎∈𝐴||𝑏 − 𝑎||) (13) For the evaluation of the diagnosis module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The specificity, sensitivity and accuracy were calculated for both 2D US image and scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The mean absolute difference (MAD) and standard deviation (SD) between results from the pseudo/reconstructed 3D volumes and results from experienced sonographers were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The metrics in verification of the system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', the stenosis grade, were compared between manual or automatic approach using the proposed technique and manual measurement using the clinical US device with the Pearson correlation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Segmentation and Diagnosis Accuracy The comparison between nine typical segmented images from U-Net and experienced sonographers was illustrated as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 7, and the images were selected from different scans at some specific locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Table I showed the average DSC and HD95 between the ground truth and prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' THE RESULTS OF VESSEL SEGMENTATION Metrics category MAB Lumen DSC 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='00% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='30% HD95(pixel) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='65 Table II showed the contingency table of the validation set of 2362 2D transverse images, and the sensitivity, specificity and accuracy were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='73, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='97 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='91 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Table III showed the diagnostic results of carotid atherosclerosis for all Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Comparison of the auto segmentation from U-net (red) and manual segmentation from ground truth (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 7 scans, and the sensitivity, specificity and accuracy of carotid atherosclerosis detection was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='71, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='85 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='80 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' THE RESULTS OF DETECTION TEST FOR 2D IMAGES Labels Predictions Positive (plaque) Negative Positive (plaque) 454 171 Negative 50 1687 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' THE DETECTION RESULTS OF CA FOR SCANS Labels Predictions Positive (plaque) Negative Positive (plaque) 25 10 Negative 10 57 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Reconstruction and Reprojection Accuracy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 8 illustrated three representative examples of the longitudinal images without regularization, with regularization clinical US images acquired by experienced sonographers, and the corresponding orientation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The results revealed that the regularized reconstructed volume was smoother with less image artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 9 demonstrated an example with large fallback of trajectory, the results showed there were still artifacts if directly apply the regularized algorithm and the proposed re-rank algorithm could remove the reconstruction artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 10 illustrated the 3D volumes reconstructed from the auto-segmentation and ground truth respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The volumes were rendered by 3D-slicer (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='slicer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The results showed that the segmentation module achieved good agreement with human label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Furthermore, the sunken of the lumen area indicated the existence of the plaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Illustration of the US longitudinal images and the corresponding orientation information from three carotid atherosclerosis patients (by rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The images in the first column were reconstructed without regularization algorithm while the images in the third column were reconstructed with regularization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The second column demonstrated the smoother results of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The fourth column represents the images acquired by sonographers using clinical US devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The images in fifth column illustrate the corresponding original position information and the images in sixth column show the regularized position information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Illustration of the proposed re-rank algorithm, the first row demonstrated the longitudinal image and corresponding position information without regularized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The second row represented the images which applied regularized algorithm and the third row showed the images which used re-rank and regularized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 8 demonstrated comparison among 5 projected images in different angles ( 𝜃 = −30°, −15°, 0°, 15°, 30° ), the image directly projected to sagittal plane and the manually acquired image by expert from the same atherosclerosis patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The results showed that the projected images in different angles could reveal more structures of the carotid than the images only projected to sagittal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' On the other hand, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 11, the image in 15° projection angle was most consistent with the clinical image obtained by expert using clinical US device, which indicated that the reprojection of 3D volume could simulate the different scan angles operated by expert to locate the best observation view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The plaque size (length and thickness) measured from the pseudo volume, reconstructed volume and images acquired by expert were compared in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The results showed good agreement between the automatic measurement from the reconstructed volume and the manual method, while the plaque size measured by pseudo volume showed large difference with the expert measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The results indicated that the 3D reconstruction could reveal the true geometry and clinical metric of the carotid artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' MAD MEASUREMENTS (N=20) BETWEEN CLINICAL US DEVICE AND THE PROPOSED TECHNIQUES Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The 3D volumes from the auto-segmentation (the first row) and ground truth (the second row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The translucent outer wall represents the vessel wall area, the inside red 3D volume represents the lumen area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The sunken of the lumen area indicated the existence of the plaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The resolution of reconstruction is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='2×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='2×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='2 𝑚𝑚3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Illustration of the projected images in different angles, from left top to right bottom were 5 projected images (𝜃 = −30°, −15°, 0°, 15°, 30° ), direct sagittal image and clinical image respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' It could be observed the sagittal image missed the part of vessel wall (in the red box) and the reprojected image with 𝜃 = 15 ° showed the most consistent structure of plague with clinical image (in the green boxes shows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 9 Plaque length(mm) Plaque thickness(mm) Plaque length (relative error) Plaque thickness (relative error 3D Reconstructed volume 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='65±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='842±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='617 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='4%± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='6% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='0%± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='2% Direct stacked Pesudo volume 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='54±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='976±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='648 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='0%± 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='0% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='4%± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='0% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Stenosis Measurement Accuracy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 12 demonstrated the linear correlation (r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='762) of the stenosis grade measured by the system and experienced sonographers using the clinical US device on 20 carotid atherosclerosis patients, which indicated the proposed technique had the strong consistency with expert manual approach in carotid atherosclerosis diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' DISCUSSION In this study, we proposed a portable freehand 3D US imaging technique for carotid artery diagnosis which could achieve real 3D geometry of carotid artery, and the method showed good agreements with manual measurement of stenosis rate and classification of diseased and healthy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The system was transportable and less dependent on operator’s experience, which make it possible for routine health check in different environments such as community or rural area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In addition, the 3D reconstructed geometry could provide visualized carotid artery structure for further atherosclerosis evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Since the large position variation or fallback movement during scan would cause reconstruction artifacts, we designed a standard scan protocol for 3D carotid US data acquisition and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The whole processing steps included automatic 3D US data acquisition, MAB and LIB segmentation, 3D reconstruction, automatic classification and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In practice, the intermediate results of each step could be reviewed and manually corrected by operator if necessary to ensure the accurate final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The diagnosis result was based on two key points: one was the accurate segmentation of vessel area, and the other was the correct reconstruction volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The segmentation determined the region of interest (ROI) used for following analysis including automatic stenosis evaluation, plaque size measurement and 3D geometry visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The wrong mask might crop regions out of the carotid artery, mislead the diagnosis network and cause confusing diagnosis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' However, if the 3D volume was directly reconstructed from original 2D frames before segmentation, the reconstruction artifacts around MAB and LIB such as misplacement or severe blurring could lead to segmentation error of vessels, especially for some cases with large position variation as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' showed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Therefore, we conducted segmentation on the original 2D US image sequence before 3D reconstruction to extract the vessel area firstly to reduce the influence of reconstruction artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' For the reconstruction process, the failure reconstruction caused by large position variation could result in severe image artifacts which totally deviated the structure of the carotid artery as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 14 For the freehand US scan, theoretically, the position information recorded by magnetic sensor would be consistent with US probe motion i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', the position of every US Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' correlation of stenosis grade between the manual measurement by expert using the clinical US device and the automatic measurement from the proposed technique on 20 carotid atherosclerosis patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Severe reconstruction artifacts caused by the large position variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The image in first row represented the reconstructed volume and the orientation information with regularized algorithm while the images in the second row represented the results without regularized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The left image shows the transverse image of the locations in the reconstructed volume marked in red boxes in the right image, the large distortion could be observed in the image while the distortion was alleviated using the regularized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Segmentation results on a transverse image collected from the 3D volume reconstructed by the original image sequence (left) and on an original transverse frame data (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' It could be observed that the severe artifacts on the left image led to wrong segmentation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='7 Stenosis grade measured by expert using clinicla US device 10 image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' However, the low precision of the sensor and inevitable hand jitter would lead to the noticeable measurement uncertainty of the position information along the scan direction and influence the reconstruction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Therefore, we adopted a novel total variation regularization algorithm to smooth the track of the position information and decrease distortion and disconnection of the image volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The position of the freehand scan can be regarded as continuous and sequential array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' therefore, the proposed regularization algorithm could reduce the uncertainty by constructing and minimize a regularized formulate in the manifold of Euclidean transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Meanwhile, a re-rank strategy was designed to solve the unordered image sequence caused by fallback movement during scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In the future, the reconstruction accuracy could be further improved using the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' After segmentation and reconstruction, the carotid artery volume could be obtained for further analysis such as healthy or diseased case diagnosis, plaque thickness, length area measurement, plaque type identification and stenosis measurement etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In the diagnosis module, the cropped and resized images instead of the whole US images were used as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Since the plaque was only located inside vessel wall area, removing useless information outside the vessel wall could accelerate network training and improve the detection accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' On the other hand, there may be low intensity area in the vessel region which could mislead the network and result in wrong classification since negative sample (no plaque) usually had low intensity in lumen area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Therefore, the MAB and LIB mask were introduced to combine the morphological information with original image information to improve the detection accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' However, the proposed approach just utilized the consecutive 2D reconstructed transverse US images to detect plaque cases, thus some cases with small plaque size or severe artifacts were wrongly classified as no plaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In the future, we will take the z axis information into account and use the whole 3D volume as input instead of detecting plaque by limited consecutive transverse slices to improve the accuracy of the diagnosis module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' We utilized a reprojection algorithm to project the carotid artery volume to longitudinal planes, so that the clinical metric such plaque length, thickness could be directly measured from the 3D volume with no need of new acquisition in sagittal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The traditional clinical carotid artery US examination required appropriate positioning and angle between transducer and neck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' which greatly relies on the operator’s experience to localize the plaque and obtain a high-quality US image,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' the proposed reprojection approach in our method was not only relatively convenient but could reveal the complete structure of the carotid artery with only one scan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' and the images obtained by our automatic method achieved great agreement with the images obtained by expert using clinical US device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In segmentation module, we used U-Net to segment the MAB and LIB in 2D US image sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Every image in the sequence was treated as a single image for the segmentation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' However, this approach didn’t exploit the context information in the adjacent frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In addition, some cases with severe noise or shadowing would result in wrong segmentation as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 15 showed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In the future, 3D convolution will be considered to correct the segmentation mistake by utilizing the context information of the adjacent frames, and sample size will be enlarged to improve the accuracy and robustness of the segmentation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' More 3D metrics such total plaque volume, vessel wall volume, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' would be evaluated for more accurate validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' On the other hand, the learning-based 3D reconstruction algorithm would be taken into account to improve the performance of reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' CONCLUSION We have proposed an automatic 3D carotid artery imaging and diagnosis technique specially designed for the portable freehand ultrasound device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The technique applied a novel 3D reconstructed algorithm and a robust segmentation algorithm for automatic carotid atherosclerosis analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The results demonstrated that the technique achieved good agreement with manual expert examination on plaque diagnosis and stenosis grade measurement, which showed the potential application on fast carotid atherosclerosis examination and the follow-ups, especially for those scenarios where professional medical device and experienced clinicians are hard to acquire such as rural area or community with large population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work was sponsored by Natural Science Foundation of China (NSFC) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='12074258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Flaherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Carotid artery stenosis as a cause of stroke,” Neuroepidemiology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 36–41, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “China cardiovascular diseases report 2018: an updated summary,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Geriatr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Cardiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' JGC, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Underhill, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Hatsukami, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fayad, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fuster, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Yuan, “MRI of carotid atherosclerosis: clinical implications and future directions,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Cardiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 165–173, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Wintermark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “High-resolution CT imaging of carotid artery atherosclerotic plaques,” Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Neuroradiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 875– 882, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Nederkoorn, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' van der Graaf, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Hunink, “Duplex ultrasound and magnetic resonance angiography compared with digital subtraction angiography in carotid artery stenosis: a systematic review,” Stroke, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1324–1331, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Inaba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Chen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Bergmann, “Carotid plaque, compared with carotid intima-media thickness, more accurately predicts coronary artery disease events: a meta-analysis,” Atherosclerosis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 220, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 128–133, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Lorenz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Markus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Bots, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Rosvall, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Sitzer, “Prediction of clinical cardiovascular events with carotid intima- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Two representative wrong segmentation examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The red line represented the automatic segmentation results by the segmentation module and the green line represented the human labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' The plaque was identified as the adventitia in the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' In the second case, the vessel wall structure was disappeared and the segmentation network resulted in wrong segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 11 media thickness: a systematic review and meta-analysis,” Circulation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 115, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 459–467, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fenster, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Parraga, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Bax, “Three-dimensional ultrasound scanning,” Interface Focus, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 503–519, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Wannarong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Progression of carotid plaque volume predicts cardiovascular events,” Stroke, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 44, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1859–1865, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fenster, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Blake, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Gyacskov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Landry, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Spence, “3D ultrasound analysis of carotid plaque volume and surface morphology,” Ultrasonics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 44, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' e153–e157, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Makris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Lavida, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Griffin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Geroulakos, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Nicolaides, “Three-dimensional ultrasound imaging for the evaluation of carotid atherosclerosis,” Atherosclerosis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 219, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 377– 383, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [12] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' AlMuhanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Carotid plaque morphometric assessment with three-dimensional ultrasound imaging,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Vasc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Surg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 61, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 690–697, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Botnar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Stuber, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Kissinger, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Kim, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Spuentrup, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Manning, “Noninvasive coronary vessel wall and plaque imaging with magnetic resonance imaging,” Circulation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 102, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 21, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2582–2587, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Herder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Johnsen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Arntzen, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Mathiesen, “Risk factors for progression of carotid intima-media thickness and total plaque area: a 13-year follow-up study: the Tromsø Study,” Stroke, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 43, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1818–1823, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Chiu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Egger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Spence, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Parraga, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fenster, “Quantification of carotid vessel wall and plaque thickness change using 3D ultrasound images,” Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3691– 3710, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [16] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Jin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' He, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Yuchi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ding, “Segmentation of the common carotid artery with active shape models from 3D ultrasound images,” in Medical Imaging 2012: Computer-Aided Diagnosis, 2012, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 8315, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 83152H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Lorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Carotid artery lumen segmentation in 3D free- hand ultrasound images using surface graph cuts,” in International conference on medical image computing and computer-assisted intervention, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 542–549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' de Ruijter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' van Sambeek, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' van de Vosse, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Lopata, “Automated 3D geometry segmentation of the healthy and diseased carotid artery in free-hand, probe tracked ultrasound images,” Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 47, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1034–1047, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Spence, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Chiu, “Segmentation of 3D ultrasound carotid vessel wall using U-Net and segmentation average network,” in 2020 42nd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2043– 2046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fenster, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Xia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Spence, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ding, “Deep learning-based carotid media-adventitia and lumen-intima boundary segmentation from three-dimensional ultrasound images,” Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3180–3193, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' De Ruijter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Muijsers, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Van de Vosse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Van Sambeek, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Lopata, “A generalized approach for automatic 3-D geometry assessment of blood vessels in transverse ultrasound images using convolutional neural networks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ultrason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ferroelectr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3326–3335, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [22] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' van Knippenberg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' van Sloun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Mischi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' de Ruijter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Lopata, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Bouwman, “Unsupervised domain adaptation method for segmenting cross-sectional CCA images,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Methods Programs Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 225, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 107037, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Azzopardi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Camilleri, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Hicks, “Bimodal automated carotid ultrasound segmentation using geometrically constrained deep neural networks,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Health Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1004–1015, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “A voxel-based fully convolution network and continuous max-flow for carotid vessel-wall-volume segmentation from 3D ultrasound images,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 39, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2844–2855, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Deep learning-based measurement of total plaque area in B-mode ultrasound images,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Health Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2967–2977, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Deep learning-based carotid plaque segmentation from B-mode ultrasound images,” Ultrasound Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 47, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2723–2733, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [27] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Xia, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Cheng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fenster, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ding, “Automatic classification of carotid ultrasound images based on convolutional neural network,” in Medical Imaging 2020: Computer-Aided Diagnosis, 2020, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 11314, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1131441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [28] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Multilevel strip pooling-based convolutional neural network for the classification of carotid plaque echogenicity,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Methods Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2021, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [29] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Shen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ding, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Xie, “NDDR-LCS: A Multi-Task Learning Method for Classification of Carotid Plaques,” in 2020 IEEE International Conference on Image Processing (ICIP), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2461–2465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [30] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Spence, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Chiu, “Three-dimensional ultrasound assessment of effects of therapies on carotid atherosclerosis using vessel wall thickness maps,” Ultrasound Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 47, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2502–2513, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [31] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fenster, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ding, “U-Net based automatic carotid plaque segmentation from 3D ultrasound images,” in Medical Imaging 2019: Computer-Aided Diagnosis, 2019, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 10950, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1119–1125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [32] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Saba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Ultrasound-based carotid stenosis measurement and risk stratification in diabetic cohort: a deep learning paradigm,” Cardiovasc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Diagn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 439, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Biswas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Two-stage artificial intelligence model for jointly measurement of atherosclerotic wall thickness and plaque burden in carotid ultrasound: A screening tool for cardiovascular/stroke risk assessment,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 123, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 103847, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [34] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “An accurate and effective FMM-based approach for freehand 3D ultrasound reconstruction,” Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 645–656, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [35] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Solberg, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Lindseth, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Torp, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Blake, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Hernes, “Freehand 3D ultrasound reconstruction algorithms—a review,” Ultrasound Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 991–1009, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [36] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “A new 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='5 D representation for lymph node detection using random sets of deep convolutional neural network observations,” in International conference on medical image computing and computer-assisted intervention, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 520–527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Lee and others, “Pseudo-label: The simple and efficient semi- supervised learning method for deep neural networks,” in Workshop on challenges in representation learning, ICML, 2013, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [38] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ronneberger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Fischer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Brox, “U-net: Convolutional networks for biomedical image segmentation,” in International Conference on Medical image computing and computer-assisted intervention, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 234–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [39] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zheng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Qian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Song, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zeng, “Improvement of 3-D Ultrasound Spine Imaging Technique Using Fast Reconstruction Algorithm,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ultrason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Ferroelectr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 3104–3113, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Chen, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zheng, “Development of Implicit Representation Method for Freehand 3D Ultrasound Image Reconstruction of Carotid Vessel,” in 2022 IEEE International Ultrasonics Symposium (IUS), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [41] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Total variation regularization of pose signals with an application to 3D freehand ultrasound,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2245–2258, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', “Deep morphology aided diagnosis network for segmentation of carotid artery vessel wall and diagnosis of carotid atherosclerosis on black-blood vessel wall MRI,” Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 5544–5561, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' [43] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Zheng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Lou, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Le, “Compact and Wireless Freehand 3D Ultrasound Real-time Spine Imaging System: A pilot study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=',” in Annual International Conference of the IEEE Engineering in Medicine and Biology Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' IEEE Engineering in Medicine and Biology Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' Annual International Conference, 2020, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} +page_content=' 2105–2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfTwM_/content/2301.03081v1.pdf'} diff --git a/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf b/89E3T4oBgHgl3EQfSAmG/content/2301.04428v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c32e048403a629b2cd37ccc40a5e4af895fc89a2 --- /dev/null +++ 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+Radosław Chrapkiewicz2, Mayukh Lahiri3, Radek Lapkiewicz1∗ +1Institute of Experimental Physics, Faculty of Physics, University of Warsaw, +ul. Pasteura 5, 02-093 Warszawa, Poland, +2CNC Program, Stanford University, Palo Alto, CA 94304, United States +3Oklahoma State University, Stillwater, OK 74078-3072, United States +∗radek.lapkiewicz@fuw.edu.pl +Interferometric methods, renowned for their reliability and precision, play a vital role +in phase imaging. Interferometry typically requires high coherence and stability be- +tween the measured and the reference beam. The presence of rapid phase fluctua- +tions averages out the interferogram, erasing the spatial phase information. This diffi- +culty can be circumvented by shortening the measurement time. However, shortening +the measurement time results in smaller photon counting rates precluding its applica- +bility to low-intensity phase imaging. We introduce and experimentally demonstrate +a phase imaging technique that is immune to position-independent, time-dependent +phase fluctuation. We accomplish this by measuring intensity correlation instead of +intensity. Our method enables using long measurement times and is therefore advan- +tageous when the photon flux is very low. We use a Fisher information-based approach +to show that the precision of phase reconstruction achieved using our method is in fact +the best achievable precision in the scenario when two photons are detected per phase +stability time. +Introduction +Phase imaging is important for applications spanning many diverse fields, including biological imaging +(1), and phase microscopy (2,3). Measurements of the phase shifts within samples can yield information +about the refractive index, thickness, and structure of an object. Interferometry (4) is a very powerful tool +that is often used in phase imaging of an object (5). Interferometric measurements allow the detection +of small variations in optical paths. There are numerous interferometric techniques such as the ones +regularly used in optical coherence tomography (6,7) or quantitative phase microscopy (8). Some of the +techniques, especially those related to biology, require very low photon fluxes. For an interferometric +measurement a wave field that has interacted with an object is superposed with a reference field and the +resulting interference pattern is detected by a camera. If the object field (probe field) and the reference +1 +arXiv:2301.11969v1 [physics.optics] 27 Jan 2023 + +field are mutually coherent, the time-averaged intensity on camera is given by (9,10): +I(x, y) = Ir + Io + 2 +� +IrIo cos[φin + φ(x, y)], +(1) +where Ir and Io are the averaged intensity of the reference and the object fields, respectively, φin is the +interferometric phase that can be changed by introducing spatial or temporal delays between the two +fields, and φ(x, y) is the phase map of the object. Standard interferometric phase imaging techniques are +based on the signature of φ(x, y) left in the detected intensity pattern. However, for any such method +to be applicable, the object field and the reference field need to be mutually coherent. Time-dependent, +uncontrollable phase fluctuations introduce incoherence between object and reference fields. The method +is therefore vulnerable to time-dependent, uncontrollable phase fluctuations that introduce incoherence +between object and reference fields. +When the phase fluctuates much faster compared to the detection time, the coherence between the +object and image fields is practically lost and, no interference will be observed, i.e., +I(x, y) = Ir + Io. +(2) +Since there is no information of φ(x, y) in this intensity pattern, the standard phase imaging scheme +becomes inapplicable to this case. One way to avoid the effect of this time-dependent phase fluctuation +is to shorten the duration of measurement (11). A short measurement time, however, reduces the amount +of detected light and is therefore impractical for imaging photo-sensitive biological specimens, which +require low-intensity light. Furthermore, for interferometric fluorescence super-resolution microscopy +(12), often very low-intensity light (13) needs to be superposed. In such cases, any time-dependent +phase fluctuations must be avoided due to the relatively long detection time requirement. +Here, we introduce a method of phase imaging that is fully resistant to time-dependent phase fluctu- +ations as long as it is possible to measure at least two photons per phase stability time. Our method is +fundamentally different from the standard phase imaging techniques (14), as we do not need interfero- +metric phase stability due to the fact that we measure intensity correlation instead of intensity. +The scheme of our experiment is illustrated in Fig. 1. A wave field that has interacted with an object +(object field) is superposed with a reference field and the resulting interference pattern is detected by +a camera. A time-dependent phase fluctuation Θ(t) is introduced in the reference field. Under these +circumstances, no information on φ(x, y) can be retrieved from the intensity pattern given by Eq. (2), +and therefore the standard phase imaging techniques become inapplicable. In the present article, we +introduce a method of phase imaging that is resistant to time-dependent phase fluctuations, provided that +phase change is uniform throughout the entire sample (15). Our method relies on measuring intensity +correlations of light and is inspired by the intensity interferometry technique introduced by Hanbury +Brown and Twiss (HBT) (16). The HBT method and its generalizations were applied to a variety of light +sources (17–25) and similarly our technique might be applied in various scenarios including laser and +thermal light as important examples. +We determine the correlation function between the intensities measured at a pair of points (x, y) and +(x′, y′) +�˜I(x, y; t)˜I(x′, y′; t) +� +∝ 1 ± 1 +2 cos [φ(x, y) − φ(x′, y′)] , +(3) +where ˜I(x, y; t) is the instantaneous intensity measured at a point (x, y) at time t. On the right hand side +2 + +Figure 1: (a) Simplified schematic of the experiment: we divide input light into two paths, an object +path(φ(x)), and a reference path. In the object path, we introduce a spatially varying phase that we want +to image. A time-fluctuating interferometric phase can be introduced to the system (Θ(t)) with no loss +in the quality of the phase retrieval. For slowly fluctuating phase Θ(t), we can measure high visibility +interference fringes (b), but no interferogram can be recorded due to insufficient photon statistics and +rapid fluctuations of (Θ(t)) - depicted in the image (c) - where fringes average to the intensity profile of +the beam having no phase information. Images (b) and (c) depict normalized one photon interference +fringes for slowly and highly fluctuating cases respectively. We also depict second-order correlation +interferograms (d) for the same photons constituting the interferograms in image (c). Even for this +highly fluctuating case, where we record only a few photons within the stability time of the phase Θ(t), +we can retrieve high visibility second-order correlation interferograms preserving full phase information +about the measured phase φ(x). +of Eq. (3), the plus (+) and minus (−) signs apply when the two points of measurement are in the same +and different beam splitter outputs, respectively. We also assume, Ir = Io. Note that the information +about the phase map of the object, which was lost in the intensity pattern [Eq. (2)], reappears in the +intensity correlation [Eq. (3)]. +3 + +%The 2nd-order intensity correlations map contains the full information required to optimally recon- +struct φ(x, y) in the extreme case when only two photons are detected during the phase stability time. +Our strategy of reconstructing the actual phase distribution in this scenario is optimal, which we prove +rigorously using estimation theory tools, namely Fisher Information and Cram´er-Rao bound (see Sup- +plementary S1 for detail). +Laser +I-sCMOS +camera +Calcite +L1 +l/2 +L2 +l/4 +Sample +j(x)/2 +l/4 +Delay line +l/4 l/2 +PBS +l/2 +Figure 2: Experimental setup for noise-resistant phase imaging. The incoming beam of Laser after pass- +ing through a λ +2 - half-wave plate, λ +4 - quarter wave plate, PBS - polarization beam splitter, and another +λ +2 plate, the beam enters a Michelson type interferometer. Each of the two paths in the interferometer is +encoded with orthogonal polarization. In one arm the spatial phase φ(x) is introduced next to the surface +of the interferometer mirror. The interferometric mirror in the other arm is given a phase fluctuation by +attaching it to a piezoelectric actuator. The two beams of the interferometric arms after combining at the +PBS pass through L1, and L2 lenses. The calcite polarizer acts as a 50/50 Beamsplitter. The I-sCMOS +- Intensified sCMOS camera records single photons at both outputs of the interferometer. The use of +short exposure time of the I-sCMOS, in the single nanosecond timescale, gives it stability and resistance +against fluctuations up to tens of MHz. +Experimental setup +The experimental setup is depicted in Fig.2. Light from a polarized, coherent source (780 nm laser) is +attenuated, spatially filtered, and directed to two arms of a polarization-based Michelson interferometer. +In order to prepare the object beam, in one of the arms, we place a phase mask to imprint spatially varying +phase φ(x) to the beam. We perform experiments with three kinds of different phase masks applied to +our object beam. We imprint a 1D quadratic local phase profile to the beam by placing a cylindrical lens +of focal length, f = 1000 mm in proximity to the mirror (Fig. 2). Additionally, we also use a spatial +light modulator (SLM) as a phase mask, as it can display any arbitrary phase profile. We imprint 1D +4 + +exponential and sinusoidal phases to our object beam by the SLM display (see supplementary S2 for +detail). +A time-dependent phase fluctuation is introduced in the other arm (the reference beam) to make +it incoherent with the object beam. This is realized with a piezoelectric actuator driven by a RAMP. +Light is combined on the PBS. Object and the reference planes are imaged onto two regions of an +Intensified sCMOS (I-sCMOS) (26) camera with a 4f system using lenses L1 and L2. After the PBS, +the object and the reference beams are distinguishable due to their orthogonal polarization. In order to +observe interference we rotate their polarization by 45 degrees with a half-wave plate and we perform the +projective measurement in the original bases with a calcite crystal. This mixes the light from both outputs +and allows us to observe interference in both outputs of the beam splitter. The visibility is reduced due +to imperfect imaging because of the path length difference in the calcite. In order to register very low +photon flux and to minimize exposure time to circumvent fluctuations, we use an Intensified-sCMOS +camera. We collect the data with a frame rate of 200 Hz. choosing a low exposure time Texp ∼ ns +allows performing measurement under phase fluctuations with frequency up to (fn ∼ 1/Texp) tens of +MHz. +Results +Data measured in our experiment consist of an average of 15 photons at both outputs of the interferome- +ter per frame. We remove temporal correlations between subsequent frames by randomly permuting the +order of frames before further processing—this process does not change the performance of our method +but allows us to simulate the conditions, in which the global phase fluctuates faster than the camera +frame rate. In such conditions, it is impossible to retrieve phases using standard interferometric methods. +Averaging recorded intensities over multiple frames or increasing measurement time would result in a +loss of the visibility of the interference fringes. In contrast, we average correlations of detected pho- +tons’ positions without any loss of the phase information. Such averaging over multiple frames results +in the reproduction of the correlation function (Eq.3), from which we can retrieve the phase profile us- +ing the standard digital holography method, Fourier off-axis holography (27). The correlation function +is measured from the coincidence map of the detected photons’ positions. This analyzing mechanism +is the essence of our noise-resistant phase imaging technique. 1D quadratic phase measurement intro- +duced by the cylindrical lens is shown in Fig. 3. The measured coincidence map (Fig. 3(a)) consists of +approximately 107 registered photon pairs with the mean number of coincidences per pixel as 100. We +estimate the phase profile shape using the collected data, and compute the Mean Squared Error (MSE) +between the measured and real value. As we show in Fig. 3(c), the MSE drops down with the total +number of measured photons, and eventually reaches the theoretical minimum, obtained with the help of +the Cram´er-Rao bound (see Supplement 2 for details). This proves, that our method of phase estimation +is optimal when at most two photons are measured during the phase stability time—notice, that this is +the most extreme limit in which one can gain any information about the phase profile. +SLM-encoded phase measurements shown in Fig. 4(a), (b), and (c) represent the measured hologram, +the retrieved phase, and the error per pixel respectively when the sinusoidal phase is applied. Similarly, +Figs. 4(e), (f), and (g) represent the measured hologram, the retrieved phase, and the error per pixel re- +spectively when the 1D exponential phase is applied. Errors in the retrieved respective phases (Fig. 4(c), +Fig. 4(g)) are due to a finite number of pixels on the SLM and discreet values of the displayed phases. +5 + +a +b +c +Figure 3: +(a) represents the measured coincidence map for a 1D quadratic phase profile, plotted with +a solid line in (b). The reconstructed phase with error bars is also shown in (b). The visibility of +the fringes in the correlation map (a) is equal to 0.62/2 (theoretical maximum with classical light is +1/2). The total number of coincidences detected in the experiment is ∼ 107. By randomly removing +a part of the collected signal, we can check how the Mean Squared Error (MSE) associated with the +phase reconstruction scales with the mean number of photons detected in one pixel during the whole +experiment (c). The MSE from the experiment is then compared with the MSE obtained using simulated +hologram, with the same parameters as in the experiment. We calculate the fundamental Cram´er-Rao +(C-R) lower bound on the MSE, assuming the visibility of hologram fringes to be equal to 0.62/2 (as in +our experiment). When no noise apart from shot noise is present (as in simulation), our method allows to +saturate this fundamental limit for large enough (∼ 5 · 104) number of photons detected per pixel. Other +possible sources of noise (e.g.) dark counts may slightly affect the MSE obtained experimentally. +Here we show that it is possible to retrieve complete phase profiles only with an average of two photons +detected per frame which is an absolute minimum of detected photons per frame. +Conclusion and Discussion +In conclusion, we demonstrate a complete retrieval of phase patterns in the presence of high-frequency +random phase fluctuations up to the order of tens of MHz when standard phase imaging techniques +fail due to the scarcity of photons within a stable phase time interval. Our method is applicable to light +sources described with different statistics, such as for example thermal light sources, and can be extended +to interference between independent sources (21,28). +6 + +Figure 4: Experimental measurement of the spatial phases with the SLM - spatial light modulator. Mea- +sured coincidence maps (correlation functions) between outputs of the interferometer for (a) sinusoidal, +and (d) exponential phases set on SLM. Each axis of coincidence maps represents the positions of pho- +tons detected along one output of the interferometer. (b) and (e) represent the aforementioned recon- +structed phases. (c) and (f) show errors and square-root of the intensities. +7 + +We want to highlight, that the presented method optimality is proven using the Cramer-Rao bound – +all the spatial phase information stored in the detected photons is retrieved (29). +High temporal resolution (short gating time) is necessary for overcoming the problem of the rapidly +fluctuating temporal phases. Such high temporal resolution in our experiment was obtained using an +image-intensified camera, which allows us to collect data with short exposure times down to a few +nanoseconds. However, our method is not limited to this camera type and can be implemented using +various high-temporal resolution detection platforms. Because of high quantum efficiency, temporal +resolution, and low noise level in recent single-photon avalanche diode (SPAD) array technology (30) +development, our method can also be implemented by SPAD arrays in the future. We stress that the tech- +nique can be implemented both in the photon counting regime and by employing less accurate intensity +measurements, yet it is the most remarkable for cases where registering more than two photons per phase +stability time is rare. Our method can be readily generalized to two-dimensional spatial phase profiles +by creating higher-dimensional correlation maps. It also allows for implementation in different degrees +of freedom, such as temporal or spectral, allowing the creation of joint probability maps both for photon +detection times or their detected wavelengths. It is also possible to incorporate an additional degree of +freedom to a measurement, measuring for instance joint temporal-spatial correlations maps. +Additionally, this method could be expanded for different situations, in which multiple photons are +detected or photons are registered at the same output. Each pair of photons can be treated as a separate +coincidence, so the number of coincidences scales with a number of detected photons n as +�n +2 +�. We can +also create such coincidence maps for multiple photons within each of the interferometer outputs as well +as between them. Such holograms can build up much faster and shorten measurement time while the +physics behind them is the same. +This method is only valid when all values of the global phase Θ have the same probability of ap- +pearing during the time interval in which the whole measurement is performed. To satisfy this condition +for arbitrary temporal phase noise, it is enough to add random uniformly distributed signal oscillating +between 0 and 2π to the unknown global phase fluctuations Θ(t). In fact, the added noise can be much +slower than the rate of phase global phase fluctuations Θ(t). +Our method opens up possible applications in wavefront sensing under low light conditions for mi- +croscopy as well as fundamental research. Unbalanced interferometers, such as ones used in the time– +bin encoding could be of particular interest, as our method enables using additional degrees of freedom +(multi-dimensional information encoding) while filtering out phase fluctuations arising, for instance, +from unmatched optical paths. In addition, because of the shorter wavelengths of X-rays (also of neu- +trons or electrons), X-ray interferometry (31,32) requires much tighter alignment and better mechanical +stability of the interferometer. We emphasize that because our technique is phase noise resistant, it holds +a potential for phase-sensitive imaging using X-ray interferometry. In addition, analogous techniques +might also find applications in matter-wave interferometry (33,34). +Acknowledgments +We acknowledge discussions with Piotr Wegrzyn, Lukasz Zinkiewicz, Michal Jachura, Wojciech Wasilewski, +and Marek Zukowski. This work was supported by the Foundation for Polish Science under the FIRST +TEAM project ’Spatiotemporal photon correlation measurements for quantum metrology and super- +resolution microscopy’ co-financed by the European Union under the European Regional Development +8 + +Fund (POIR.04.04.00-00-3004/17-00), and by the National Laboratory for Photonics and Quantum Tech- +nologies—NLPQT (POIR.04.02.00.00-B003/18). +Supplementary materials +S1 - Fundamental precision limits of phase imaging with fluctuating reference arm +S2 - Experimental setup details +References +1. Y. Park, C. Depeursinge, G. Popescu, Nature Photonics 12, 578 (2018). +2. G. Popescu, T. Ikeda, R. R. Dasari, M. S. Feld, Optics Letters 31, 775 (2006). +3. Z. Wang, et al., Optics Express 19, 1016 (2011). +4. P. Hariharan, Ed. 2, Optical Interferometry (Academic Press, 2003). +5. T. Ikeda, G. Popescu, R. R. Dasari, M. S. Feld, Optics Letters 30, 1165 (2005). +6. D. Huang, et al., Science 254, 1178 (1991). +7. M. Sticker, C. K. Hitzenberger, R. Leitgeb, A. F. Fercher, Optics Letters 26, 518 (2001). +8. E. Cuche, F. Bevilacqua, C. Depeursinge, Optics Letters 24, 291 (1999). +9. E. Hecht, Ed. 5, Chapter 9, Optics (Pearson Education Limited, 2017). +10. G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill, New York, 2011). +11. G. Magyar, L. Mandel, Nature 198, 255 (1963). +12. L. A. Rozema, et al., Phys. Rev. Lett. 112, 223602 (2014). +13. P. A. Morris, R. S. Aspden, J. E. C. Bell, R. W. Boyd, M. J. Padgett, Nature Communications 6 +(2015). +14. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, 1971). +15. J. Szuniewicz, et al., Rochester Conference on Coherence and Quantum Optics (CQO-11) (OSA, +2019). +16. R. Hanbury Brown, R. Q. Twiss, Nature 177, 27 (1956). +17. C. K. Hong, Z. Y. Ou, L. Mandel, Physical Review Letters 59, 2044 (1987). +18. Z. Y. Ou, E. C. Gage, B. E. Magill, L. Mandel, Journal of the Optical Society of America B 6, 100 +(1989). +9 + +19. T. B. Pittman, J. D. Franson, Physical Review Letters 90 (2003). +20. R. L. Pfleegor, L. Mandel, Physical Review 159, 1084 (1967). +21. J. G. Rarity, P. R. Tapster, R. Loudon, Journal of Optics B: Quantum and Semiclassical Optics 7, +S171 (2005). +22. X. Li, L. Yang, L. Cui, Z. Y. Ou, D. Yu, Optics Express 16, 12505 (2008). +23. A. J. Bennett, R. B. Patel, C. A. Nicoll, D. A. Ritchie, A. J. Shields, Nature Physics 5, 715 (2009). +24. Y.-S. Kim, O. Slattery, P. S. Kuo, X. Tang, Physical Review A 87 (2013). +25. R. Chrapkiewicz, M. Jachura, K. Banaszek, W. Wasilewski, Nature Photonics 10, 576 (2016). +26. R. Chrapkiewicz, W. Wasilewski, K. Banaszek, Optics Letters 39, 5090 (2014). +27. J. Mertz, Introduction to Optical Microscopy (Roberts and Company Publishers, 2009.). +28. H. Paul, Rev. Mod. Phys. 58, 209 (1986). +29. H. Cram´er, Mathematical methods of statistics, vol. 26 (Princeton university press, 1999). +30. I. M. Antolovic, C. Bruschini, E. Charbon, Optics Express 26, 22234 (2018). +31. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, C. David, Physical Review Letters 105 (2010). +32. T. Weitkamp, B. N¨ohammer, A. Diaz, C. David, E. Ziegler, Applied Physics Letters 86, 054101 +(2005). +33. E. M. Rasel, M. K. Oberthaler, H. Batelaan, J. Schmiedmayer, A. Zeilinger, Physical Review Letters +75, 2633 (1995). +34. M. Arndt, A. Ekers, W. von Klitzing, H. Ulbricht, New Journal of Physics 14, 125006 (2012). +10 + +Supplementary materials and methods +1 +S1: Fundamental precision limits of phase imaging with +fluctuating reference arm +1.1 +The measurement model +Two cameras are set on the two outputs of the interferometer, each of them consists of +the same number of pixels npix. The sample area giving the additional phase φi is imaged +to the pixel number i on both cameras. Only two photons are received per the stability +time of the interferometer phase. A single measurement consists of a detection of these +two photons. The output of the single measurement is a pair (i+/−, j+/−). Numbers i, j +stand for the numbers of pixels in which photons were detected, whereas indices + or +− indicates in which of the two outputs the corresponding photon was measured. The +probability of measuring a single photon in a pixel i+/− is: +p(i+/−) = ˜NIi +1 +2(1 ± v cos(φi + θ)), +(1) +where ˜N is a normalization factor, v is interferometer visibility, θ is an extra, global, +fluctuating phase, and Ii is the intensity of the illuminating the phase mask in the are +corresponding to pixel i. Phase θ is stable during the detection of each photon pair, +its value for each pair is independently drawn from the continuous uniform probability +distribution U(0, 2π). There is no information about θ value in each experiment, so the +observed probability of obtaining pair (i+/−, j+/−) in every single frame is: +p(i+/−, j+/−) = +� 2π +0 +p(i+/−, j+/−, θ)dθ +(2) +p(i+/−, j+/−, θ) is a joint probability distribution of measuring pair (i+/−, j+/−) with the +fixed value of θ. From equation 2 we obtain the formulas: +p(i+, j+) = p(i−, j−) = NIiIj(2 + v2 cos(φi − φj)) +(3) +p(i+, j−) = p(i−, j+) = NIiIj(2 − v2 cos(φi − φj)) +(4) +N is a new normalization factor. The above equations are our starting point to further +inference about the maximal precision of the measurement. Full information about each +single measurement is included in the dependendence of the probability p of the specific +result of a measurement (i±, j±) on the estimated parameters φi. +1 +arXiv:2301.11969v1 [physics.optics] 27 Jan 2023 + +1.2 +Cramér-Rao bound +In order to calculate maximal precision of estimation of the parameters φi, Fisher Infor- +mation (FI) matrix will be calculated. There are 4 different types of events, which can +occur during one experiment - two photons may be detected in one output (+ or −) or +in different outputs ( we distinguish between +− and −+). We can distinguish between +these 4 types, so the FI is the sum of FI matrices for all events’ types: +Ftot = F++ + F−− + F+− + F−+ +(5) +From equations 3 and 4 we can simply conclude, that F++ = F−− and F+− = F−+. In +the next part of the article F++ matrix will be calculated. +In order to simplify the formulas, the following notation will be used: +p(i+, j+) ≡ p(i, j), +∂ +∂φk +≡ ∂k, +F ≡ F++ +The element of the FI matrix can be written in the following form: +Fkl = +npix +� +i,j=1 +∂kp(i, j)∂lp(i, j) +p(i, j) +, +(6) +Subsequently: +∂kp(i, j) = NIiIjv2(δjk − δik) sin(φi − φj) +(7) +∂kp(i, j)∂lp(i, j) = (δjk − δik)(δjl − δil)N2I2 +i I2 +j v4 sin2(φi − φj) +(8) +Consequently, the matrix element is: +Fkl = +npix +� +i,j=1 +(δjk − δik)(δjl − δil)NIiIjv4 sin2(φi − φj) +2 + v2 cos(φi − φj) +(9) +If k ̸= l, then for any m we have δmkδml = 0, so (δjk − δik)(δjl − δil) = −δjkδil − δikδjl. +That means, that non-diagonal matrix elements are: +Fkl = −2NIkIlv4 sin2(φk − φl) +2 + v2 cos(φk − φl) +, +k ̸= l +(10) +With the help of the equality (δjk − δik)2 = δjk + δik − 2δikδjk we can obtain diagonal +terms of F: +Fkk = 2NIkv4 +npix +� +i=1 +Ii sin2(φi − φk) +2 + v2 cos(φi − φk) +(11) +For any function f: +npix +� +i=1 +f(φi, Ii) = npix⟨f(φi, Ii)⟩i, +(12) +2 + +where ⟨f(φi, Ii)⟩i is the mean value of the function over all pixels. In the next steps, +the number of pixels is assumed to be big and each phase in the sample occurs with the +same frequency. What is more, intensity of illuminating beam Ii is assumed to change +slowly compared to the change of phase φi. In other words, many different phases occur +in the region with approximately constant intensity. From these assumptions we obtain +the equality: +⟨f(φi, Ii)⟩i = 1 +2π +� 2π +0 +f(φ, ⟨I⟩)dφ, +(13) +where ⟨I⟩ stands for the mean intensity of the illuminating beam. +Using the above +assumptions, we can rewrite equation 11 as: +Fkk = 2NIk⟨I⟩v4 npix +2π +� 2π +0 +sin2(φ − φk) +2 + v2 cos(φ − φk)dφ +(14) +Consequently all diagonal terms of F are the same: +Fkk = 2N⟨I⟩Iknpix(2 − +� +4 − v4) +(15) +Now we need to calculate the value of a normalization factor N. We will use the fact, +that sum of propabilities of all events must be equal to one: +npix +� +i,j=1 +p(i+, j+) + p(i+, j−) + p(i−, j+) + p(i−, j−) = 1 +(16) +Using equations 3 and 4 we obtain: +8N +npix +� +i,j=1 +IiIj = 1 +(17) +We can rewrite the sum in the above equation as: +npix +� +i,j=1 +IiIj = +�npix +� +i=1 +Ii +�2 += n2 +pix⟨I⟩2 +(18) +and obtain: +N = +1 +8n2 +pix⟨I⟩2 +(19) +Finally F++ matrix can be written in the form: +Fkl = +� +� +� +� +� +� +� +1 +4npix +Ik +⟨I⟩(2 − +√ +4 − v4) +for k = l +− +1 +4n2 +pix +IkIl +⟨I⟩2 +2v4 sin2(φk−φl) +2+v2 cos(φk−φl) +for k ̸= l +(20) +3 + +We have calculated F++ matrix, which is obviously similar to F−− matrix, because +formulas for propabilities in both cases are the same. Analogous calculation show, that +also F+− = F−+ = F++. Using the FI additivity we obtain the terms of Ftot matrix: +Ftot = 4F++ +(21) +This is the FI matrix for a single measurement. If the whole experiment consists of nmes +independent repetitions of the single measurement, we obtain the FI: +F (nmes) +tot += 4nmesF++ = 2nphotF++, +(22) +where nphot stands for the total number of measured photons during the experiment. In +the next part F stands for the whole FI associated with detection of nphot number of +photons. Terms of this matrix are: +Fkl = +� +� +� +� +� +� +� +nphot +npix +Ik +⟨I⟩(1 − +� +1 − v4/4) +for k = l +− nphot +2n2 +pix +IkIl +⟨I⟩2 +2v4 sin2(φk−φl) +2+v2 cos(φk−φl) +for k ̸= l +(23) +From the Cramer-Rao bound, the minimal possible variance for estimating φk satisfy +the inequality: +∆2φk ≥ (F −1)kk +(24) +In general, the estimator which satisfy the above inequality may not exist, however, it is +possible to get arbitrary close to the above bound if the number of measurement is big +enough. That means, that the inequality becomes an equality if nphot → ∞. To simplify +the calculations we also use the inequality: +(F −1)kk ≥ (Fkk)−1, +(25) +which is true for all hermitian F. It’s clear, that in the general case the above inequality is +not saturable. However, in our case the non-diagonal terms are asymptotically npix times +smaller than diagonal terms. npix is also size of the F matrix. It may be proven, that for +such scaling of non-diagonal terms with the size of matrix, the above inequality becomes +saturable for npix → ∞. Using both of above inequalities, we obtain the following bound: +∆φk ≥ +� +npix⟨I⟩ +nphotIk +1 +� +1 − +� +1 − v4/4 +(26) +The value nk = nphotIk +npix⟨I⟩ may be interpreted as the expected value of photons detected in +pixel number k ( in any output). The above bound may be rewritten in the intuitive +form: +∆φk ≥ +� +1 +nk +1 +� +1 − +� +1 − v4/4 +(27) +From this form of the inequality it’s clear, that the accuracy of measuring the value of the +particular phase depends directly on the numer of photons interacting with the measured +area. +4 + +1.3 +Comparison with long-stability-time interferometer +Let’s compare our result with the phase estimation precision limit for an interferometer +with slowly fluctuating phase θ. First of all, let’s notice that we can’t beat the accuracy +achievable in the situation, in which extra phase θ is known for all the detected photons. +Indeed, the information we get in a situation with unknown θ is always smaller, even +if the stability time if the interferometer is bigger. If θ values are known, each single +photon detection could be treated as an independent event (which was not the case in +the previous section). Let’s calculate the FI matrix for the single photon detection when +θ is fixed. Single measurement is fully described by the probability distribution from +equation 1. Further we obtain: +∂kp(i+/−) = ∓1 +2δki ˜NIiv sin(φi + θ) +(28) +In this case FI matrix has the form: +Fkl = +npix +� +i=1 +∂kp(i+)∂lp(i+) +p(i+) ++ +npix +� +i=1 +∂kp(i−)∂lp(i−) +p(i−) +(29) +From equation 28 it’s clear, that all non-diagonal terms of the F matrix are equal to +zero. This is because we obtain information about the φi phase only in case of detection +a photon in the pixel i+/−. The diagonal terms are: +Fkk = ˜NIi +v2 sin2(φi + θ) +1 − v2 cos2(φi + θ) +(30) +To make this case similar to the case descriped in the previous section let’s assume, that +θ fluctuates and each value of θ appears with the same frequency ( the difference is that θ +fluctuates slowly and we know it’s value). Then the mean FI for the single measurement +is: +⟨Fkk⟩θ = 1 +2π +� 2π +0 +Fkkdθ = +Ii +npix⟨I⟩ +� +1 − +� +1 − v2 +� +, +(31) +where formula ˜N = +1 +npix⟨I⟩ obtained from the normalization condition was used. If nmes +measurements were made, nphot photons were consumed. If we define nk = nphotIk +npix⟨I⟩ as in +the previous section, we obtain the best possible accuracy of measuring each phase φk: +∆φk ≥ +� +1 +nk +1 +� +1 − +√ +1 − v2 +(32) +Equation 32 is very similar to the equation 27- the only difference is that term v4 +4 is +substitude by the term v2. That means, that having only two photons per phase fluc- +tuations stability time, leads to decrease of the effective visibility of the interferometer +from v to v2 +2 . As it was mentioned, it’s not possible to beat the bound from equation +32 if θ value is not known in each measurement, even if the number of detected photons +5 + +in a phase stability time was increased. However, we can get close to that bound, if the +phase stability time is big enough. Indeed, if we can measure many photons, when θ is +stable, we don’t really need to care about its unknown value and obtain relative values +of φk using the same method as in case of known θ ( it might be assumed to equal 0). +This scheme is repeated independently for each θ . The bound from the equation 30 is +saturated, because the number of measurements is big enough. That means, that we can +also saturate the bound resulting from the mean FI (equation 32). +2 +S2: Experimental setup details +This is a polarization-based Michelson interferometer. As a light source, we use a diode +laser at a wavelength of 780 nm coupled to a single-mode fiber. At the output of the +fiber, for polarization control, the attenuated beam passes through a half-wave plate, a +quarter-wave plate, and polarizing beam splitter (PBS), and another half-wave plate, and +then enters a Michelson-type interferometer. Each of the two paths in the interferometer +is encoded with orthogonal polarization. In order to prepare the object beam, in one +of the arms of the interferometer, we build two kinds of slightly modified setups - one +with a cylindrical lens placed in front of one of the mirror in the horizontally polarized +light beam path in the Michelson interferometer while in the other setup we replace +the mirror in the same path with a spatial light modulator (SLM), thereby introducing +spatially varying phase φ(x) onto the beam in that path. In one arm the spatial phase +φ(x) is introduced next to the surface of the interferometer mirror. The interferometeric +mirror in the other arm is given a phase fluctuation by attaching it to a piezoelectric +actuator. +We perform experiments with three kinds of different phase masks applied to our +object beam. Our first configuration is to imprint a one dimensional quadratic local phase +profile to the beam by placing a cylindrical lens of focal length, f = 1000 mm in proximity +to the mirror (Fig. 2 in the main text). Additionally, in our second configuration with +SLM (from the HOLOEYE PLUTO) as a phase mask, we can display any arbitrary phase +profile. As an example, we imprint one dimensional exponential and sinusoidal phases +to our object beam by the SLM display. +We introduce a time-dependent phase fluctuation is in the other arm (the reference +beam - vertically polarized beam path in the interferometer) to make it incoherent with +the object beam. This is realized with a piezoelectric actuator driven by a RAMP of 1.234 +Hz. This shouldn’t be confused with the maximal noise frequency for which our method +works. Both of the object and reference beams are combined on the polarizing beam +splitter (PBS). Afterthat, they are imaged onto two regions of an Intensified sCMOS +(I-sCMOS - with the image intensifier from Hamamatsu V7090D-71-G272 and sCMOS +from Andor Zyla) camera with a 4f system using lenses L3 and L4 of focal length 200 mm. +To observe the interference, the orthogonally polarized object and the reference beam +are required to be indistinguishable, and to do so, we rotate the polarization of both +beams by 45 degrees with a half-wave plate and we perform projective measurement in +the original bases with a calcite crystal. Here, the calcite acts as a 50/50 Beamsplitter. +6 + +I-sCMOS +Camera +Calcite +λ/2 +λ/4 +Laser +PBS +λ/2 +L2 +PH +M +L1 +λ/4 +λ/4 +PBS +λ/2 +Delay line +M +SLM +L3 +L4 +M +M +Figure 1: +Experimental setup for noise-resistant phase imaging. The incoming beam +of Laser after passing through a λ/2 - half-wave plate, λ/4 - quarter wave plate, PBS +- polarization beam splitter, and another λ/2 plate, the beam enters a Michelson type +interferometer. Each of the two paths in the interferometer is encoded with orthogonal +polarization. In one arm the spatial phase φ(x) is introduced by the spatial light modu- +lator (SLM). The interferometric mirror in the other arm is given a phase fluctuation by +attaching it to a piezoelectric actuator. The two beams of the interferometric arms after +combining at the PBS pass through L3, and L4 lenses. The calcite polarizer acts as a +50/50 Beamsplitter. The I-sCMOS - Intensified sCMOS camera records single photons +at both outputs of the interferometer. The use of short exposure time of the I-sCMOS, +in the single nanosecond timescale, gives it stability and resistance against fluctuations +up to tens of MHz.. +This mixes the light from both outputs and allows us to observe interference in both +outputs of the splitter. The I-sCMOS camera records single photons at both outputs +of the interferometer. The use of short exposure time of the I-sCMOS, in the single +nanosecond timescale, gives it stability and resistance against fluctuations up to tens of +MHz. We collect the data with 200 Hz of frame rate. +7 + diff --git a/8dFLT4oBgHgl3EQfBS4r/content/tmp_files/load_file.txt b/8dFLT4oBgHgl3EQfBS4r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd19c56af1e9dcba96cb3415fe4cdb500ae4dca8 --- /dev/null +++ b/8dFLT4oBgHgl3EQfBS4r/content/tmp_files/load_file.txt @@ -0,0 +1,513 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf,len=512 +page_content='Noise Resistant Phase Imaging with Intensity Correlation Jerzy Szuniewicz1, Stanisław Kurdziałek1, Sanjukta Kundu1, Wojciech Zwolinski1, Radosław Chrapkiewicz2, Mayukh Lahiri3, Radek Lapkiewicz1∗ 1Institute of Experimental Physics, Faculty of Physics, University of Warsaw, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Pasteura 5, 02-093 Warszawa, Poland, 2CNC Program, Stanford University, Palo Alto, CA 94304, United States 3Oklahoma State University, Stillwater, OK 74078-3072, United States ∗radek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='lapkiewicz@fuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='pl Interferometric methods, renowned for their reliability and precision, play a vital role in phase imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Interferometry typically requires high coherence and stability be- tween the measured and the reference beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The presence of rapid phase fluctua- tions averages out the interferogram, erasing the spatial phase information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This diffi- culty can be circumvented by shortening the measurement time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' However, shortening the measurement time results in smaller photon counting rates precluding its applica- bility to low-intensity phase imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We introduce and experimentally demonstrate a phase imaging technique that is immune to position-independent, time-dependent phase fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We accomplish this by measuring intensity correlation instead of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Our method enables using long measurement times and is therefore advan- tageous when the photon flux is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We use a Fisher information-based approach to show that the precision of phase reconstruction achieved using our method is in fact the best achievable precision in the scenario when two photons are detected per phase stability time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Introduction Phase imaging is important for applications spanning many diverse fields, including biological imaging (1), and phase microscopy (2,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Measurements of the phase shifts within samples can yield information about the refractive index, thickness, and structure of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Interferometry (4) is a very powerful tool that is often used in phase imaging of an object (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Interferometric measurements allow the detection of small variations in optical paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' There are numerous interferometric techniques such as the ones regularly used in optical coherence tomography (6,7) or quantitative phase microscopy (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Some of the techniques, especially those related to biology, require very low photon fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' For an interferometric measurement a wave field that has interacted with an object is superposed with a reference field and the resulting interference pattern is detected by a camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' If the object field (probe field) and the reference 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='11969v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='optics] 27 Jan 2023 field are mutually coherent, the time-averaged intensity on camera is given by (9,10): I(x, y) = Ir + Io + 2 � IrIo cos[φin + φ(x, y)], (1) where Ir and Io are the averaged intensity of the reference and the object fields, respectively, φin is the interferometric phase that can be changed by introducing spatial or temporal delays between the two fields, and φ(x, y) is the phase map of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Standard interferometric phase imaging techniques are based on the signature of φ(x, y) left in the detected intensity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' However, for any such method to be applicable, the object field and the reference field need to be mutually coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Time-dependent, uncontrollable phase fluctuations introduce incoherence between object and reference fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The method is therefore vulnerable to time-dependent, uncontrollable phase fluctuations that introduce incoherence between object and reference fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' When the phase fluctuates much faster compared to the detection time, the coherence between the object and image fields is practically lost and, no interference will be observed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=', I(x, y) = Ir + Io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' (2) Since there is no information of φ(x, y) in this intensity pattern, the standard phase imaging scheme becomes inapplicable to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' One way to avoid the effect of this time-dependent phase fluctuation is to shorten the duration of measurement (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A short measurement time, however, reduces the amount of detected light and is therefore impractical for imaging photo-sensitive biological specimens, which require low-intensity light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Furthermore, for interferometric fluorescence super-resolution microscopy (12), often very low-intensity light (13) needs to be superposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In such cases, any time-dependent phase fluctuations must be avoided due to the relatively long detection time requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Here, we introduce a method of phase imaging that is fully resistant to time-dependent phase fluctu- ations as long as it is possible to measure at least two photons per phase stability time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Our method is fundamentally different from the standard phase imaging techniques (14), as we do not need interfero- metric phase stability due to the fact that we measure intensity correlation instead of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The scheme of our experiment is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A wave field that has interacted with an object (object field) is superposed with a reference field and the resulting interference pattern is detected by a camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A time-dependent phase fluctuation Θ(t) is introduced in the reference field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Under these circumstances, no information on φ(x, y) can be retrieved from the intensity pattern given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' (2), and therefore the standard phase imaging techniques become inapplicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In the present article, we introduce a method of phase imaging that is resistant to time-dependent phase fluctuations, provided that phase change is uniform throughout the entire sample (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Our method relies on measuring intensity correlations of light and is inspired by the intensity interferometry technique introduced by Hanbury Brown and Twiss (HBT) (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The HBT method and its generalizations were applied to a variety of light sources (17–25) and similarly our technique might be applied in various scenarios including laser and thermal light as important examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We determine the correlation function between the intensities measured at a pair of points (x, y) and (x′, y′) �˜I(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' t)˜I(x′, y′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' t) � ∝ 1 ± 1 2 cos [φ(x, y) − φ(x′, y′)] , (3) where ˜I(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' t) is the instantaneous intensity measured at a point (x, y) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' On the right hand side 2 Figure 1: (a) Simplified schematic of the experiment: we divide input light into two paths, an object path(φ(x)), and a reference path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In the object path, we introduce a spatially varying phase that we want to image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A time-fluctuating interferometric phase can be introduced to the system (Θ(t)) with no loss in the quality of the phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' For slowly fluctuating phase Θ(t), we can measure high visibility interference fringes (b), but no interferogram can be recorded due to insufficient photon statistics and rapid fluctuations of (Θ(t)) - depicted in the image (c) - where fringes average to the intensity profile of the beam having no phase information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Images (b) and (c) depict normalized one photon interference fringes for slowly and highly fluctuating cases respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We also depict second-order correlation interferograms (d) for the same photons constituting the interferograms in image (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Even for this highly fluctuating case, where we record only a few photons within the stability time of the phase Θ(t), we can retrieve high visibility second-order correlation interferograms preserving full phase information about the measured phase φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' (3), the plus (+) and minus (−) signs apply when the two points of measurement are in the same and different beam splitter outputs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We also assume, Ir = Io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Note that the information about the phase map of the object, which was lost in the intensity pattern [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' (2)], reappears in the intensity correlation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' (3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 3 %The 2nd-order intensity correlations map contains the full information required to optimally recon- struct φ(x, y) in the extreme case when only two photons are detected during the phase stability time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Our strategy of reconstructing the actual phase distribution in this scenario is optimal, which we prove rigorously using estimation theory tools, namely Fisher Information and Cram´er-Rao bound (see Sup- plementary S1 for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Laser I-sCMOS camera Calcite L1 l/2 L2 l/4 Sample j(x)/2 l/4 Delay line l/4 l/2 PBS l/2 Figure 2: Experimental setup for noise-resistant phase imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The incoming beam of Laser after pass- ing through a λ 2 - half-wave plate, λ 4 - quarter wave plate, PBS - polarization beam splitter, and another λ 2 plate, the beam enters a Michelson type interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Each of the two paths in the interferometer is encoded with orthogonal polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In one arm the spatial phase φ(x) is introduced next to the surface of the interferometer mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The interferometric mirror in the other arm is given a phase fluctuation by attaching it to a piezoelectric actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The two beams of the interferometric arms after combining at the PBS pass through L1, and L2 lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The calcite polarizer acts as a 50/50 Beamsplitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The I-sCMOS Intensified sCMOS camera records single photons at both outputs of the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The use of short exposure time of the I-sCMOS, in the single nanosecond timescale, gives it stability and resistance against fluctuations up to tens of MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Experimental setup The experimental setup is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Light from a polarized, coherent source (780 nm laser) is attenuated, spatially filtered, and directed to two arms of a polarization-based Michelson interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In order to prepare the object beam, in one of the arms, we place a phase mask to imprint spatially varying phase φ(x) to the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We perform experiments with three kinds of different phase masks applied to our object beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We imprint a 1D quadratic local phase profile to the beam by placing a cylindrical lens of focal length, f = 1000 mm in proximity to the mirror (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Additionally, we also use a spatial light modulator (SLM) as a phase mask, as it can display any arbitrary phase profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We imprint 1D 4 exponential and sinusoidal phases to our object beam by the SLM display (see supplementary S2 for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A time-dependent phase fluctuation is introduced in the other arm (the reference beam) to make it incoherent with the object beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This is realized with a piezoelectric actuator driven by a RAMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Light is combined on the PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Object and the reference planes are imaged onto two regions of an Intensified sCMOS (I-sCMOS) (26) camera with a 4f system using lenses L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' After the PBS, the object and the reference beams are distinguishable due to their orthogonal polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In order to observe interference we rotate their polarization by 45 degrees with a half-wave plate and we perform the projective measurement in the original bases with a calcite crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This mixes the light from both outputs and allows us to observe interference in both outputs of the beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The visibility is reduced due to imperfect imaging because of the path length difference in the calcite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In order to register very low photon flux and to minimize exposure time to circumvent fluctuations, we use an Intensified-sCMOS camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We collect the data with a frame rate of 200 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' choosing a low exposure time Texp ∼ ns allows performing measurement under phase fluctuations with frequency up to (fn ∼ 1/Texp) tens of MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Results Data measured in our experiment consist of an average of 15 photons at both outputs of the interferome- ter per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We remove temporal correlations between subsequent frames by randomly permuting the order of frames before further processing—this process does not change the performance of our method but allows us to simulate the conditions, in which the global phase fluctuates faster than the camera frame rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In such conditions, it is impossible to retrieve phases using standard interferometric methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Averaging recorded intensities over multiple frames or increasing measurement time would result in a loss of the visibility of the interference fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In contrast, we average correlations of detected pho- tons’ positions without any loss of the phase information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Such averaging over multiple frames results in the reproduction of the correlation function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='3), from which we can retrieve the phase profile us- ing the standard digital holography method, Fourier off-axis holography (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The correlation function is measured from the coincidence map of the detected photons’ positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This analyzing mechanism is the essence of our noise-resistant phase imaging technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 1D quadratic phase measurement intro- duced by the cylindrical lens is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The measured coincidence map (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 3(a)) consists of approximately 107 registered photon pairs with the mean number of coincidences per pixel as 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We estimate the phase profile shape using the collected data, and compute the Mean Squared Error (MSE) between the measured and real value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' As we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 3(c), the MSE drops down with the total number of measured photons, and eventually reaches the theoretical minimum, obtained with the help of the Cram´er-Rao bound (see Supplement 2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This proves, that our method of phase estimation is optimal when at most two photons are measured during the phase stability time—notice, that this is the most extreme limit in which one can gain any information about the phase profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' SLM-encoded phase measurements shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 4(a), (b), and (c) represent the measured hologram, the retrieved phase, and the error per pixel respectively when the sinusoidal phase is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Similarly, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 4(e), (f), and (g) represent the measured hologram, the retrieved phase, and the error per pixel re- spectively when the 1D exponential phase is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Errors in the retrieved respective phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 4(c), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 4(g)) are due to a finite number of pixels on the SLM and discreet values of the displayed phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 5 a b c Figure 3: (a) represents the measured coincidence map for a 1D quadratic phase profile, plotted with a solid line in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The reconstructed phase with error bars is also shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The visibility of the fringes in the correlation map (a) is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='62/2 (theoretical maximum with classical light is 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The total number of coincidences detected in the experiment is ∼ 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' By randomly removing a part of the collected signal, we can check how the Mean Squared Error (MSE) associated with the phase reconstruction scales with the mean number of photons detected in one pixel during the whole experiment (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The MSE from the experiment is then compared with the MSE obtained using simulated hologram, with the same parameters as in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We calculate the fundamental Cram´er-Rao (C-R) lower bound on the MSE, assuming the visibility of hologram fringes to be equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='62/2 (as in our experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' When no noise apart from shot noise is present (as in simulation), our method allows to saturate this fundamental limit for large enough (∼ 5 · 104) number of photons detected per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Other possible sources of noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=') dark counts may slightly affect the MSE obtained experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Here we show that it is possible to retrieve complete phase profiles only with an average of two photons detected per frame which is an absolute minimum of detected photons per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Conclusion and Discussion In conclusion, we demonstrate a complete retrieval of phase patterns in the presence of high-frequency random phase fluctuations up to the order of tens of MHz when standard phase imaging techniques fail due to the scarcity of photons within a stable phase time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Our method is applicable to light sources described with different statistics, such as for example thermal light sources, and can be extended to interference between independent sources (21,28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 6 Figure 4: Experimental measurement of the spatial phases with the SLM - spatial light modulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Mea- sured coincidence maps (correlation functions) between outputs of the interferometer for (a) sinusoidal, and (d) exponential phases set on SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Each axis of coincidence maps represents the positions of pho- tons detected along one output of the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' (b) and (e) represent the aforementioned recon- structed phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' (c) and (f) show errors and square-root of the intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 7 We want to highlight, that the presented method optimality is proven using the Cramer-Rao bound – all the spatial phase information stored in the detected photons is retrieved (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' High temporal resolution (short gating time) is necessary for overcoming the problem of the rapidly fluctuating temporal phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Such high temporal resolution in our experiment was obtained using an image-intensified camera, which allows us to collect data with short exposure times down to a few nanoseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' However, our method is not limited to this camera type and can be implemented using various high-temporal resolution detection platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Because of high quantum efficiency, temporal resolution, and low noise level in recent single-photon avalanche diode (SPAD) array technology (30) development, our method can also be implemented by SPAD arrays in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We stress that the tech- nique can be implemented both in the photon counting regime and by employing less accurate intensity measurements, yet it is the most remarkable for cases where registering more than two photons per phase stability time is rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Our method can be readily generalized to two-dimensional spatial phase profiles by creating higher-dimensional correlation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' It also allows for implementation in different degrees of freedom, such as temporal or spectral, allowing the creation of joint probability maps both for photon detection times or their detected wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' It is also possible to incorporate an additional degree of freedom to a measurement, measuring for instance joint temporal-spatial correlations maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Additionally, this method could be expanded for different situations, in which multiple photons are detected or photons are registered at the same output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Each pair of photons can be treated as a separate coincidence, so the number of coincidences scales with a number of detected photons n as �n 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We can also create such coincidence maps for multiple photons within each of the interferometer outputs as well as between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Such holograms can build up much faster and shorten measurement time while the physics behind them is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This method is only valid when all values of the global phase Θ have the same probability of ap- pearing during the time interval in which the whole measurement is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' To satisfy this condition for arbitrary temporal phase noise, it is enough to add random uniformly distributed signal oscillating between 0 and 2π to the unknown global phase fluctuations Θ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In fact, the added noise can be much slower than the rate of phase global phase fluctuations Θ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Our method opens up possible applications in wavefront sensing under low light conditions for mi- croscopy as well as fundamental research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Unbalanced interferometers, such as ones used in the time– bin encoding could be of particular interest, as our method enables using additional degrees of freedom (multi-dimensional information encoding) while filtering out phase fluctuations arising, for instance, from unmatched optical paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In addition, because of the shorter wavelengths of X-rays (also of neu- trons or electrons), X-ray interferometry (31,32) requires much tighter alignment and better mechanical stability of the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We emphasize that because our technique is phase noise resistant, it holds a potential for phase-sensitive imaging using X-ray interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In addition, analogous techniques might also find applications in matter-wave interferometry (33,34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Acknowledgments We acknowledge discussions with Piotr Wegrzyn, Lukasz Zinkiewicz, Michal Jachura, Wojciech Wasilewski, and Marek Zukowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This work was supported by the Foundation for Polish Science under the FIRST TEAM project ’Spatiotemporal photon correlation measurements for quantum metrology and super- resolution microscopy’ co-financed by the European Union under the European Regional Development 8 Fund (POIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='00-00-3004/17-00), and by the National Laboratory for Photonics and Quantum Tech- nologies—NLPQT (POIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='00-B003/18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Supplementary materials S1 - Fundamental precision limits of phase imaging with fluctuating reference arm S2 - Experimental setup details References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Park, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Depeursinge, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Popescu, Nature Photonics 12, 578 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Popescu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Ikeda, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Dasari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Feld, Optics Letters 31, 775 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Wang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=', Optics Express 19, 1016 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Hariharan, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 2, Optical Interferometry (Academic Press, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Ikeda, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Popescu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Dasari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Feld, Optics Letters 30, 1165 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Huang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=', Science 254, 1178 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Sticker, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Hitzenberger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Leitgeb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Fercher, Optics Letters 26, 518 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Cuche, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Bevilacqua, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Depeursinge, Optics Letters 24, 291 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Hecht, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 5, Chapter 9, Optics (Pearson Education Limited, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill, New York, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Magyar, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Mandel, Nature 198, 255 (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Rozema, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 112, 223602 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Morris, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Aspden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Bell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Boyd, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Padgett, Nature Communications 6 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Collier, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Burckhardt, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Lin, Optical Holography (Academic, 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Szuniewicz, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=', Rochester Conference on Coherence and Quantum Optics (CQO-11) (OSA, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Hanbury Brown, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Twiss, Nature 177, 27 (1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Hong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Ou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Mandel, Physical Review Letters 59, 2044 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Ou, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Gage, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Magill, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Mandel, Journal of the Optical Society of America B 6, 100 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Pittman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Franson, Physical Review Letters 90 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Pfleegor, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Mandel, Physical Review 159, 1084 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Rarity, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Tapster, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Loudon, Journal of Optics B: Quantum and Semiclassical Optics 7, S171 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Cui, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Ou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Yu, Optics Express 16, 12505 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Bennett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Patel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Nicoll, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Ritchie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Shields, Nature Physics 5, 715 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Kim, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Slattery, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Kuo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Tang, Physical Review A 87 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Chrapkiewicz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Jachura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Banaszek, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Wasilewski, Nature Photonics 10, 576 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Chrapkiewicz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Wasilewski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Banaszek, Optics Letters 39, 5090 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Mertz, Introduction to Optical Microscopy (Roberts and Company Publishers, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Paul, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 58, 209 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Cram´er, Mathematical methods of statistics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 26 (Princeton university press, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Antolovic, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Bruschini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Charbon, Optics Express 26, 22234 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Zanette, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Weitkamp, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Donath, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Rutishauser, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' David, Physical Review Letters 105 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Weitkamp, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' N¨ohammer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Diaz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' David, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Ziegler, Applied Physics Letters 86, 054101 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Rasel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Oberthaler, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Batelaan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Schmiedmayer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Zeilinger, Physical Review Letters 75, 2633 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Arndt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Ekers, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' von Klitzing, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Ulbricht, New Journal of Physics 14, 125006 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 10 Supplementary materials and methods 1 S1: Fundamental precision limits of phase imaging with fluctuating reference arm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='1 The measurement model Two cameras are set on the two outputs of the interferometer, each of them consists of the same number of pixels npix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The sample area giving the additional phase φi is imaged to the pixel number i on both cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Only two photons are received per the stability time of the interferometer phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' A single measurement consists of a detection of these two photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The output of the single measurement is a pair (i+/−, j+/−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Numbers i, j stand for the numbers of pixels in which photons were detected, whereas indices + or − indicates in which of the two outputs the corresponding photon was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The probability of measuring a single photon in a pixel i+/− is: p(i+/−) = ˜NIi 1 2(1 ± v cos(φi + θ)), (1) where ˜N is a normalization factor, v is interferometer visibility, θ is an extra, global, fluctuating phase, and Ii is the intensity of the illuminating the phase mask in the are corresponding to pixel i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Phase θ is stable during the detection of each photon pair, its value for each pair is independently drawn from the continuous uniform probability distribution U(0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' There is no information about θ value in each experiment, so the observed probability of obtaining pair (i+/−, j+/−) in every single frame is: p(i+/−, j+/−) = � 2π 0 p(i+/−, j+/−, θ)dθ (2) p(i+/−, j+/−, θ) is a joint probability distribution of measuring pair (i+/−, j+/−) with the fixed value of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' From equation 2 we obtain the formulas: p(i+, j+) = p(i−, j−) = NIiIj(2 + v2 cos(φi − φj)) (3) p(i+, j−) = p(i−, j+) = NIiIj(2 − v2 cos(φi − φj)) (4) N is a new normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The above equations are our starting point to further inference about the maximal precision of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Full information about each single measurement is included in the dependendence of the probability p of the specific result of a measurement (i±, j±) on the estimated parameters φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='11969v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='optics] 27 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='2 Cramér-Rao bound In order to calculate maximal precision of estimation of the parameters φi, Fisher Infor- mation (FI) matrix will be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' There are 4 different types of events, which can occur during one experiment - two photons may be detected in one output (+ or −) or in different outputs ( we distinguish between +− and −+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We can distinguish between these 4 types, so the FI is the sum of FI matrices for all events’ types: Ftot = F++ + F−− + F+− + F−+ (5) From equations 3 and 4 we can simply conclude, that F++ = F−− and F+− = F−+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In the next part of the article F++ matrix will be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In order to simplify the formulas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' the following notation will be used: p(i+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j+) ≡ p(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' ∂ ∂φk ≡ ∂k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' F ≡ F++ The element of the FI matrix can be written in the following form: Fkl = npix � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='j=1 ∂kp(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j)∂lp(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j) p(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' (6) Subsequently: ∂kp(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j) = NIiIjv2(δjk − δik) sin(φi − φj) (7) ∂kp(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j)∂lp(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j) = (δjk − δik)(δjl − δil)N2I2 i I2 j v4 sin2(φi − φj) (8) Consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' the matrix element is: Fkl = npix � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='j=1 (δjk − δik)(δjl − δil)NIiIjv4 sin2(φi − φj) 2 + v2 cos(φi − φj) (9) If k ̸= l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' then for any m we have δmkδml = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' so (δjk − δik)(δjl − δil) = −δjkδil − δikδjl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' That means, that non-diagonal matrix elements are: Fkl = −2NIkIlv4 sin2(φk − φl) 2 + v2 cos(φk − φl) , k ̸= l (10) With the help of the equality (δjk − δik)2 = δjk + δik − 2δikδjk we can obtain diagonal terms of F: Fkk = 2NIkv4 npix � i=1 Ii sin2(φi − φk) 2 + v2 cos(φi − φk) (11) For any function f: npix � i=1 f(φi, Ii) = npix⟨f(φi, Ii)⟩i, (12) 2 where ⟨f(φi, Ii)⟩i is the mean value of the function over all pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In the next steps, the number of pixels is assumed to be big and each phase in the sample occurs with the same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' What is more, intensity of illuminating beam Ii is assumed to change slowly compared to the change of phase φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In other words, many different phases occur in the region with approximately constant intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' From these assumptions we obtain the equality: ⟨f(φi, Ii)⟩i = 1 2π � 2π 0 f(φ, ⟨I⟩)dφ, (13) where ⟨I⟩ stands for the mean intensity of the illuminating beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Using the above assumptions, we can rewrite equation 11 as: Fkk = 2NIk⟨I⟩v4 npix 2π � 2π 0 sin2(φ − φk) 2 + v2 cos(φ − φk)dφ (14) Consequently all diagonal terms of F are the same: Fkk = 2N⟨I⟩Iknpix(2 − � 4 − v4) (15) Now we need to calculate the value of a normalization factor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We will use the fact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' that sum of propabilities of all events must be equal to one: npix � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='j=1 p(i+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j+) + p(i+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j−) + p(i−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j+) + p(i−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' j−) = 1 (16) Using equations 3 and 4 we obtain: 8N npix � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='j=1 IiIj = 1 (17) We can rewrite the sum in the above equation as: npix � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='j=1 IiIj = �npix � i=1 Ii �2 = n2 pix⟨I⟩2 (18) and obtain: N = 1 8n2 pix⟨I⟩2 (19) Finally F++ matrix can be written in the form: Fkl = � � � � � � � 1 4npix Ik ⟨I⟩(2 − √ 4 − v4) for k = l − 1 4n2 pix IkIl ⟨I⟩2 2v4 sin2(φk−φl) 2+v2 cos(φk−φl) for k ̸= l (20) 3 We have calculated F++ matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' which is obviously similar to F−− matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' because formulas for propabilities in both cases are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Analogous calculation show, that also F+− = F−+ = F++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Using the FI additivity we obtain the terms of Ftot matrix: Ftot = 4F++ (21) This is the FI matrix for a single measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' If the whole experiment consists of nmes independent repetitions of the single measurement, we obtain the FI: F (nmes) tot = 4nmesF++ = 2nphotF++, (22) where nphot stands for the total number of measured photons during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In the next part F stands for the whole FI associated with detection of nphot number of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Terms of this matrix are: Fkl = � � � � � � � nphot npix Ik ⟨I⟩(1 − � 1 − v4/4) for k = l − nphot 2n2 pix IkIl ⟨I⟩2 2v4 sin2(φk−φl) 2+v2 cos(φk−φl) for k ̸= l (23) From the Cramer-Rao bound,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' the minimal possible variance for estimating φk satisfy the inequality: ∆2φk ≥ (F −1)kk (24) In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' the estimator which satisfy the above inequality may not exist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' it is possible to get arbitrary close to the above bound if the number of measurement is big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' That means, that the inequality becomes an equality if nphot → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' To simplify the calculations we also use the inequality: (F −1)kk ≥ (Fkk)−1, (25) which is true for all hermitian F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' It’s clear, that in the general case the above inequality is not saturable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' However, in our case the non-diagonal terms are asymptotically npix times smaller than diagonal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' npix is also size of the F matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' It may be proven, that for such scaling of non-diagonal terms with the size of matrix, the above inequality becomes saturable for npix → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Using both of above inequalities, we obtain the following bound: ∆φk ≥ � npix⟨I⟩ nphotIk 1 � 1 − � 1 − v4/4 (26) The value nk = nphotIk npix⟨I⟩ may be interpreted as the expected value of photons detected in pixel number k ( in any output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The above bound may be rewritten in the intuitive form: ∆φk ≥ � 1 nk 1 � 1 − � 1 − v4/4 (27) From this form of the inequality it’s clear, that the accuracy of measuring the value of the particular phase depends directly on the numer of photons interacting with the measured area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='3 Comparison with long-stability-time interferometer Let’s compare our result with the phase estimation precision limit for an interferometer with slowly fluctuating phase θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' First of all, let’s notice that we can’t beat the accuracy achievable in the situation, in which extra phase θ is known for all the detected photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Indeed, the information we get in a situation with unknown θ is always smaller, even if the stability time if the interferometer is bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' If θ values are known, each single photon detection could be treated as an independent event (which was not the case in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Let’s calculate the FI matrix for the single photon detection when θ is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Single measurement is fully described by the probability distribution from equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Further we obtain: ∂kp(i+/−) = ∓1 2δki ˜NIiv sin(φi + θ) (28) In this case FI matrix has the form: Fkl = npix � i=1 ∂kp(i+)∂lp(i+) p(i+) + npix � i=1 ∂kp(i−)∂lp(i−) p(i−) (29) From equation 28 it’s clear, that all non-diagonal terms of the F matrix are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This is because we obtain information about the φi phase only in case of detection a photon in the pixel i+/−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The diagonal terms are: Fkk = ˜NIi v2 sin2(φi + θ) 1 − v2 cos2(φi + θ) (30) To make this case similar to the case descriped in the previous section let’s assume, that θ fluctuates and each value of θ appears with the same frequency ( the difference is that θ fluctuates slowly and we know it’s value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Then the mean FI for the single measurement is: ⟨Fkk⟩θ = 1 2π � 2π 0 Fkkdθ = Ii npix⟨I⟩ � 1 − � 1 − v2 � , (31) where formula ˜N = 1 npix⟨I⟩ obtained from the normalization condition was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' If nmes measurements were made, nphot photons were consumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' If we define nk = nphotIk npix⟨I⟩ as in the previous section, we obtain the best possible accuracy of measuring each phase φk: ∆φk ≥ � 1 nk 1 � 1 − √ 1 − v2 (32) Equation 32 is very similar to the equation 27- the only difference is that term v4 4 is substitude by the term v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' That means, that having only two photons per phase fluc- tuations stability time, leads to decrease of the effective visibility of the interferometer from v to v2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' As it was mentioned, it’s not possible to beat the bound from equation 32 if θ value is not known in each measurement, even if the number of detected photons 5 in a phase stability time was increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' However, we can get close to that bound, if the phase stability time is big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Indeed, if we can measure many photons, when θ is stable, we don’t really need to care about its unknown value and obtain relative values of φk using the same method as in case of known θ ( it might be assumed to equal 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This scheme is repeated independently for each θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The bound from the equation 30 is saturated, because the number of measurements is big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' That means, that we can also saturate the bound resulting from the mean FI (equation 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 2 S2: Experimental setup details This is a polarization-based Michelson interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' As a light source, we use a diode laser at a wavelength of 780 nm coupled to a single-mode fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' At the output of the fiber, for polarization control, the attenuated beam passes through a half-wave plate, a quarter-wave plate, and polarizing beam splitter (PBS), and another half-wave plate, and then enters a Michelson-type interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Each of the two paths in the interferometer is encoded with orthogonal polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In order to prepare the object beam, in one of the arms of the interferometer, we build two kinds of slightly modified setups - one with a cylindrical lens placed in front of one of the mirror in the horizontally polarized light beam path in the Michelson interferometer while in the other setup we replace the mirror in the same path with a spatial light modulator (SLM), thereby introducing spatially varying phase φ(x) onto the beam in that path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In one arm the spatial phase φ(x) is introduced next to the surface of the interferometer mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The interferometeric mirror in the other arm is given a phase fluctuation by attaching it to a piezoelectric actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We perform experiments with three kinds of different phase masks applied to our object beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Our first configuration is to imprint a one dimensional quadratic local phase profile to the beam by placing a cylindrical lens of focal length, f = 1000 mm in proximity to the mirror (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 2 in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Additionally, in our second configuration with SLM (from the HOLOEYE PLUTO) as a phase mask, we can display any arbitrary phase profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' As an example, we imprint one dimensional exponential and sinusoidal phases to our object beam by the SLM display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We introduce a time-dependent phase fluctuation is in the other arm (the reference beam - vertically polarized beam path in the interferometer) to make it incoherent with the object beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This is realized with a piezoelectric actuator driven by a RAMP of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='234 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' This shouldn’t be confused with the maximal noise frequency for which our method works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Both of the object and reference beams are combined on the polarizing beam splitter (PBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Afterthat, they are imaged onto two regions of an Intensified sCMOS (I-sCMOS - with the image intensifier from Hamamatsu V7090D-71-G272 and sCMOS from Andor Zyla) camera with a 4f system using lenses L3 and L4 of focal length 200 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' To observe the interference, the orthogonally polarized object and the reference beam are required to be indistinguishable, and to do so, we rotate the polarization of both beams by 45 degrees with a half-wave plate and we perform projective measurement in the original bases with a calcite crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Here, the calcite acts as a 50/50 Beamsplitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 6 I-sCMOS Camera Calcite λ/2 λ/4 Laser PBS λ/2 L2 PH M L1 λ/4 λ/4 PBS λ/2 Delay line M SLM L3 L4 M M Figure 1: Experimental setup for noise-resistant phase imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The incoming beam of Laser after passing through a λ/2 - half-wave plate, λ/4 - quarter wave plate, PBS polarization beam splitter, and another λ/2 plate, the beam enters a Michelson type interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' Each of the two paths in the interferometer is encoded with orthogonal polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' In one arm the spatial phase φ(x) is introduced by the spatial light modu- lator (SLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The interferometric mirror in the other arm is given a phase fluctuation by attaching it to a piezoelectric actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The two beams of the interferometric arms after combining at the PBS pass through L3, and L4 lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The calcite polarizer acts as a 50/50 Beamsplitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The I-sCMOS - Intensified sCMOS camera records single photons at both outputs of the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The use of short exposure time of the I-sCMOS, in the single nanosecond timescale, gives it stability and resistance against fluctuations up to tens of MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content='. This mixes the light from both outputs and allows us to observe interference in both outputs of the splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The I-sCMOS camera records single photons at both outputs of the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' The use of short exposure time of the I-sCMOS, in the single nanosecond timescale, gives it stability and resistance against fluctuations up to tens of MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' We collect the data with 200 Hz of frame rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} +page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFLT4oBgHgl3EQfBS4r/content/2301.11969v1.pdf'} diff --git a/A9AzT4oBgHgl3EQf__9t/content/tmp_files/2301.01956v1.pdf.txt b/A9AzT4oBgHgl3EQf__9t/content/tmp_files/2301.01956v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3c4a438d3ce30524555c82d16e388b3c5fe4fcd6 --- /dev/null +++ b/A9AzT4oBgHgl3EQf__9t/content/tmp_files/2301.01956v1.pdf.txt @@ -0,0 +1,1542 @@ +High-level semantic feature matters few-shot unsupervised domain adaptation +Lei Yu1, Wanqi Yang1*, Shengqi Huang1, Lei Wang2, Ming Yang1 +1School of Computer and Electronic Information, Nanjing Normal University, China +2School of Computing and Information Technology, University of Wollongong, Australia +yulei@njnu.edu.cn, yangwq@njnu.edu.cn, huangshengqi@njnu.edu.cn, leiw@uow.edu.au, myang@njnu.edu.cn +Abstract +In few-shot unsupervised domain adaptation (FS-UDA), most +existing methods followed the few-shot learning (FSL) meth- +ods to leverage the low-level local features (learned from con- +ventional convolutional models, e.g., ResNet) for classifica- +tion. However, the goal of FS-UDA and FSL are relevant yet +distinct, since FS-UDA aims to classify the samples in target +domain rather than source domain. We found that the local +features are insufficient to FS-UDA, which could introduce +noise or bias against classification, and not be used to effec- +tively align the domains. To address the above issues, we aim +to refine the local features to be more discriminative and rele- +vant to classification. Thus, we propose a novel task-specific +semantic feature learning method (TSECS) for FS-UDA. +TSECS learns high-level semantic features for image-to-class +similarity measurement. Based on the high-level features, we +design a cross-domain self-training strategy to leverage the +few labeled samples in source domain to build the classi- +fier in target domain. In addition, we minimize the KL diver- +gence of the high-level feature distributions between source +and target domains to shorten the distance of the samples be- +tween the two domains. Extensive experiments on Domain- +Net show that the proposed method significantly outperforms +SOTA methods in FS-UDA by a large margin (i.e., ∼ 10%). +keywords +Few-shot unsupervised domain adaptation, image-to-class +similarity, high-level semantic features, cross-domain self- +training, cross-attention. +Introduction +Currently, a setting namely few-shot unsupervised domain +adaptation (FS-UDA) (Huang et al. 2021)(Yang et al. 2022), +which utilizes few labeled data in source domain to train +a model to classify unlabeled data in target domain, owns +its potential feasibility. Typically, a FS-UDA model could +learn general knowledge from base classes during training +to guide classification in novel classes during testing. It is +known that both insufficient labels in source domain and +large domain shift make FS-UDA as a challenging task. +Previous studies, e.g., IMSE (Huang et al. 2021), first fol- +lowed several few-shot learning (FSL) methods (Li et al. +*The corresponding author is Wanqi Yang. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +Figure 1: A 5-way 1-shot task for FS-UDA where the sup- +port set includes five classes and one sample for each class. +The figure shows the similarity of query images to every +support classes and the spatial similarity of query images +to the predicted support class. We found using local fea- +tures could cause some inaccurate regions of query images +to match the incorrect classes, while our semantic features +make the object region in query images similar with their +true class, thus achieving correct classification. +2019)(Tzeng et al. 2017) to learn the local features by us- +ing convolutional models (e.g., ResNet) and then leveraged +them to learn image-to-class similarity pattern for classifica- +tion. However, we wish to clarify that the goal of FS-UDA +and FSL are relevant yet distinct, since both of them suf- +fer from insufficient labeled training data whereas FS-UDA +aims to classify the samples in target domain rather than +source domain. As shown in Fig. 1, by visualizing the spatial +similarity of query images to predicted support classes, we +found using local features causes the inaccurate regions of +query images to match incorrect classes. This reason might +be that few labeled samples and large domain shift between +the support and query sets simultaneously result in the con- +ventional local features in FSL to fail in classification. In this +sense, the local features are insufficient to FS-UDA, which +could introduce noise or bias against the classification in tar- +get domain and not be used to effectively align the domains. +To address this issue, we aim to refine the low-level local +arXiv:2301.01956v1 [cs.CV] 5 Jan 2023 + +support set in the source domain (sketch) +sailboat +bed +glasses +television +snowman +query set in the target domain (clipart) +local features +semantic features (ours) +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +bed +local features +semantic features (ours) +0.6 +0.6 +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +television +as +saFigure 2: Illustration of the process for cross-domain self- +training in TSECS. Different shapes represent different do- +mains. We first select the ‘confidence’ target samples (e.g., +a) that are very similar to support classes, and then regard +them as the new class prototypes to further classify the other +target samples (e.g., b, c). This process is executed itera- +tively with using class matching loss to narrow the distance +of query images and their most similar support classes. +features to be more discriminative and relevant to classifica- +tion, i.e., high-level semantic features, and meanwhile align +the semantic features for domain adaptation. Therefore, +we propose a novel task-specific semantic feature method +(TSECS) that learns the semantic features for each task by +clustering the local features of support set and query set. To +obtain the related semantics from previous tasks, the cluster +centroids of the current task are then fused by cross-attention +with that of the previous task to generate high-level semantic +features to boost classification performance. +Moreover, for the domain shift between source and tar- +get domains, many domain adaptation methods (Saito et al. +2018)(Tzeng et al. 2017)(Tzeng et al. 2014) reduced the dis- +tribution discrepancy between domains by using a discrim- +inator to adverse against feature embedding. However, this +way could fail in aligning the samples of the same class be- +tween domains due to label missing in target domain, which +could make the classes of two domains mismatched and thus +affect the classification. Therefore, we aim to align the high- +level semantic features by minimizing the KL divergence +of the semantic feature distributions between domains, and +meanwhile design a cross-domain self-training strategy to +train the classifier in target domain. +We hypothesis that there are usually several ‘confidence’ +samples in target domain that could be classified correctly by +support set in source domain, in other words, they are very +similar to their class prototypes in source domain. Mean- +while, the target domain samples in the same class are more +similar to each other than that of other classes. Based on this, +we regard these ‘confidence’ samples in the target domain as +new prototypes of the classes, which replace those from the +support set of source domain. As shown in Fig. 2, several +‘confidence’ samples (e.g., a) can be selected as prototypes +of their similar classes for classification (e.g., b and c) in tar- +get domain. Moreover, the process is conducted iteratively +by using class matching loss for better domain alignment. +In sum, we propose the novel method, namely TSECS, +for FS-UDA. It refines the local features of convolutional +network to generate specific semantic features of each task, +and meanwhile perform cross-domain self-training to trans- +port labels from support set in the source domain to query +set in the target domain to effectively classify the samples in +target domain. Our contributions can be summarized as: +(1) A novel solution for FS-UDA. TSECS aims to learn +high-level semantic features for classification and do- +main alignment, which could be regarded as a more ef- +fective and efficient way than using local features. +(2) Task-specific semantic embedding for few-shot set- +ting. It can be seamlessly add to existing FSL/FS-UDA +models, which could alleviate the bias of classification. +(3) Cross-domain self-training for domain alignment. It +is designed to bring the samples of the same class close, +which could guide effective domain alignment. +We conduct extensive experiments on DomainNet. Our +method significantly outperforms SOTA methods in FS- +UDA by a large margin up to ∼ 10%. +Related Works +Unsupervised domain adaptation. The conventional UDA +methods aim to reduce discrepancy between source domain +and target domain in the feature space and utilize suffi- +ciently labeled source domain data to classify data from tar- +get domain. The difference between unsupervised domain +adaptation methods often lies in the evaluation of domain +discrepancy and the objective function of model training. +Several researchers (Long et al. 2015)(Tzeng et al. 2014) +minimize the feature discrepancy by using maximum mean +discrepancy to measure the discrepancy between the distri- +bution of domains. Moreover, adversarial training (Tzeng +et al. 2017)(Ganin et al. 2016) to learn domain-invariant fea- +tures is usually used to tackle domain shift. Several meth- +ods (Tang, Chen, and Jia 2020)(Zou et al. 2019)(Zou et al. +2018)(Kim et al. 2021)train the classifier in both source do- +main and target domain and utilize pseudo-labels from target +domain to calculate classification loss. Overall, these UDA +methods all require sufficiently labeled source domain data +to realize domain alignment and classification, but they per- +form poor when labeled source domain data are insufficient. +Few-shot learning. Few-shot learning has two main +streams, metric-based and optimization-based approaches. +Optimization-based methods (Bertinetto et al. 2019)(Finn, +Abbeel, and Levine 2017)(Ravi and Larochelle 2017) usu- +ally train a meta learner over auxiliary dataset to learn +a general initialization model, which can fine-tune and +adapt to new tasks very soon. The main purpose of metric- +based methods (Li et al. 2019)(Snell, Swersky, and Zemel +2017)(Vinyals et al. 2016)(Ye et al. 2020) is that learn a gen- +eralizable feature embedding for metric learning, which can +immediately adapt to new tasks without any fine-tune and +retraining. Typically, ProtoNet (Snell, Swersky, and Zemel +2017) learns the class prototypes in the support set and clas- +sifies the query images based on the maximum similarity +to these prototypes. Other than these metric-based methods +on feature maps, many methods on local features have ap- +peared. DN4 (Li et al. 2019) utilizes large amount of local +features to measure the similarity between support and query + +select 'confidence" sanples +use new prototypes for +as new prototypes +classification in target domain +O +b +O +0 +00 +00 +lass natching loss +0 +00 +dims prototypes +doeifiad query imega +[sonmce dm ngin) +(trt domain) +at din) +(trt domain)sets instead of flattening the feature map into a long vec- +tor. Based on local features, DeepEMD (Zhang et al. 2020) +adopts Earth Mover’s Distance distance to measure the re- +lationship between query and support sets. Furthermore, a +few recent works focus on the issue of cross-domain FSL in +which domain shift exists between data of meta tasks and +new tasks. The baseline models (Chen et al. 2019) are used +to do cross-domain FSL. LFT (Tseng et al. 2020) performs +adaptive feature transformation to tackle the domain shift. +Few-shot unsupervised domain adaptation. Compared +with UDA, FS-UDA is to deal with many UDA tasks by +leveraging few labeled source domain samples for each. And +compared with cross-domain FSL, FS-UDA are capable of +handling the circumstances of no available labels in the tar- +get domain, and large domain gap between the support and +query sets in every task. For the one-shot UDA (Luo et al. +2020), it deals with the case that only one unlabeled target +sample is available, but does not require the source domain +to be few-shot, which is different from ours. Recently, there +are a few attempts in FS-UDA. PCS (Yue et al. 2021) per- +forms prototype self-supervised learning in cross-domain, +but they require enough unlabeled source samples to learn +prototypes and ignore task-level transfer, which is also dif- +ferent from ours. meta-FUDA (Yang et al. 2022) lever- +ages meta learning-based optimization to perform task-level +transfer and domain-level transfer jointly. IMSE (Huang +et al. 2021) utilizes local features to learn similarity patterns +for cross-domain similarity measurement. However, they did +not consider that local features could bring the noise or bias +to affect classification and domain alignment. Thus, we pro- +pose task-specific semantic features to solve this problem. +Methodology +Problem Definition +A N-way, K-shot FS-UDA task. Table 1 shows the main +symbols used in this paper. The FS-UDA setting includes +two domains: a source domain S and a target domain T. +A N-way, K-shot FS-UDA task includes a support set XS +from S and a query set QT from T. The support set XS +contains N classes and K samples per class in the source +domain. The query set QT contains the same N classes as +in XS and Nq target domain samples per class. To classify +query images in QT to the correct class in XS, it is popular +to train a general model from base classes to adapt to handle +new N-way, K-shot FS-UDA tasks for testing. +Auxiliary dataset and episodic training. As in (Huang +et al. 2021), the base classes are collected from an auxil- +iary dataset Daux to perform episodic training to learn the +general model. Note that the base classes in Daux are com- +pletely different from new classes in testing tasks, which are +unseen during episodic training. Moreover, Daux includes +labeled source domain data and unlabeled target domain +data for FS-UDA. We construct large amounts of episodes, +each containing {XS, QS, QT } as in (Huang et al. 2021), to +simulate the testing tasks for task-level generalization. Note +that QS is introduced into episodic training to calculate clas- +sification loss and perform domain alignment with QT . +The flowchart of our method. Fig. 3 illustrates our +Table 1: Notations +Notations +Descriptions +N ∈ R +The number of classes in the task. +K ∈ R +The number of samples per class in support set. +XS, QS, QT +Support set of source domain, and query sets +of source domain and target domain. +H, W, d ∈ R +The height, width, and channel of feature map. +L ∈ RHW ×d +The local feature vectors in the feature map. +k ∈ R +The number of semantic clusters for an episode. +C ∈ Rk×d +The centroids of the clusters. +F, ˆF, +The semantic feature map, semantic features and +ˆFXS, ˆFQS, ˆFQT +the parts of support and query sets in both domains. +M c +q ∈ RH×W ×N +The 3-D similarity matrix for classification. +pc +q ∈ RKHW +Similarity pattern vectors of a query image q +pi +q ∈ RHW +with a support class c and a support image i, +ppos +q +, pneg +q +∈ RKHW +and the most similar class and the second one for q. +µA, µB ∈ RHW ×d +The mean of semantic features or similarity patterns. +ΣA, ΣB ∈ RHW ×HW +Covariance matrix of semantic features +or similarity patterns. +λsfa, λspa, λclm +Weight parameters of three loss terms in Eq. (6). +method for 5-way, 1-shot FS-UDA tasks. In each episode, +a support set (XS) and two query sets (QS and QT ) are first +through the convolution network (e.g., ResNet) to extract +their local features. Then, the task-specific semantic embed- +ding module refines the local features to generate semantic +features, which is computational efficient due to dimension +reduction. Also, based on semantic features of QS and QT , +we leverage their similarity patterns (Huang et al. 2021) to +calculate image-to-class similarity for classification with the +loss Lcls. To improve its performance, cross-domain self- +training module is performed to introduce the class proto- +types of target domain and train a target domain classifier +with a class matching loss Lclm. In addition, the seman- +tic features and similarity patterns from both domains are +further aligned by calculating their alignment losses Lsfa +and Lspa, respectively. Finally, the losses above are back- +propagated to update our model. After episodic training over +all episodes, we utilize the learned model to test new FS- +UDA tasks. Then, we calculate the averaged classification +accuracy on these tasks for performance evaluation. +Task-specific Semantic Feature Learning +Most FSL methods and FS-UDA methods learned local fea- +tures from convolutional networks for classification. How- +ever, we found that the local features could introduce noise +or bias that is valid for classification and domain alignment. +Thus, we aim to refine the local features to generate high- +level semantic features for each task. In the following, we +will introduce our semantic feature embedding module. +First of all, in each episode, all local features L +∈ +R(|XS|+|QS|+|QT |)HW ×d are extracted from the convolu- +tional network, where | · | is the number of samples in a +set. Then, we cluster the local features to generate different +semantic clusters for support set and query set, respectively, +since clustering the two sets together could result in the clus- +ters that relate to the domains due to the presence of large do- +main gap. For simplification, we adopt K-means for cluster- +ing, and meanwhile utilize the singular value decomposition +(SVD) to adaptively take the number of eigenvalues greater +than a certain threshold as the cluster number k (k ≪ d) for +each task. Afterwards, we calculate the task-specific seman- + +Figure 3: Illustration of our method training per episode for 1-shot FS-UDA tasks. First, support classes and query images +from both domains are through a convolution network to extract their local features, followed by the task-specific semantic +embedding module to learn high-level semantic features. Then, these semantic features are fed into the cross-domain self- +training module to update the class prototypes for target domain classification and calculate the class matching loss Lclm. +Meanwhile, these semantic features are also used to generate similarity patterns in IMSE (Huang et al. 2021) for classification +loss Lcls. In addition, both semantic features and similarity patterns from both domains are aligned by the domain alignment +module with the alignment losses Lsfa and Lspa, respectively. Finally, all the losses are backpropagated to update our model. +tic feature map F ∈ R(|XS|+|QS|+|QT |)HW ×k by measuring +the Cosine similarity between the local features L and the +centroids C ∈ Rk×d of all semantic clusters, i.e., F = +L +||L||2 · +C⊤ +||C||2 . Finally, we split F to 2×2 blocks based on height and +weight dimension of the feature map, and then concatenate +the four blocks together along the channel to generate se- +mantic features ˆF ∈ R +1 +4 (|XS|+|QS|+|QT |)HW ×4k. This is a +simple yet effective way to maintain discriminative ability +and spatial information of semantic features. +Moreover, to leverage the semantics from previous tasks +to guide the semantic feature learning of the current task, we +utilize the centroids of previous clusters to update the initial- +ization of clustering centroids by cross-attention (Li et al. +2020). This makes K-means clustering converge rapidly. +After obtaining the semantic features ˆF, we use them for +domain alignment and classification. Firstly, ˆF is partitioned +into ˆFXS, ˆFQS, ˆFQT along with the first dimension. Then, +we align ˆFQS and ˆFQT by minimizing the KL divergence of +their distributions that will be introduced later. Meanwhile, +we utilize ˆFXS, ˆFQS and ˆFQT to build 3-D similarity matrix +M c +q (Huang et al. 2021) between support and query sets. Fi- +nally, we calculate the similarity pattern pc +q (measuring the +similarity between query sample q and support class c) for +classification (Huang et al. 2021). The classification loss us- +ing cross-entropy can be written by: +Lcls = − +1 +|QS| +� +q∈QS +log( +exp(1 · pc +q) +�K +i=1 exp(1 · piq) +) +(1) +Cross-domain Self-training +Since there is large domain shift between source and target +domains, as well as label missing in target domain, adver- +sarial domain adaptation on low-level local features cannot +make samples of the same class between domains close, and +thus could make the classes of two domains mismatched. +To alleviate the mismatching issue, we aim to find the +most similar ‘confidence’ samples in QT with XS to guide +classification in target domain. We assume that it usually +exists that the ‘confidence’ samples in QT could be clas- +sified correctly by XS, when the distributions between do- +mains are aligned. We iteratively select the ‘confidence’ +samples in QT as the new prototypes to replace that in XS +for classification, as shown in Fig. 2. We call the process as +cross-domain self-training. The process can find more ‘con- +fidence’ samples from QT than that in XS for the same +class, which could correct some misclassified samples in +QT , thereby lightening the impact of domain gap. +Moreover, to improve the performance of the target do- +main classifier, we aim to make target domain samples q +in QT closer to their most similar class and meanwhile far +away from the other classes. Thus, we first calculate its sim- +ilarity patterns ppos +q +(with the most similar class) and pneg +q +(with the second similar class), and then design the class +matching loss with a margin m, which can be written by +Lclm = +� +q∈QT +max(softmax(pneg +q +)−softmax(ppos +q +)+m, 0), +(2) +where the similarity to the most similar class should be +greater by m than the second similar class. +Two-level Domain Alignment +Conventional +adversarial +domain +adaptation +methods +(Ganin et al. 2016)(Tzeng et al. 2017) iteratively train a +discriminator to align the distribution of domains by adver- +sarial training among tasks. However, they cannot be used +to align the semantic features, because our semantic features +are relevant to tasks, the semantics of the same channel + +Task-specific semantic embedding +Local features +Semantic feature maps +High-level semantic features + Support class +(Source domain) +Similarity patterns in IMSE +Qurey image +MH +(Target domain) +conv +Lcls +Classification loss +Query image +(Source domain) +Split into 2 x 2 blocks +I Update the class + and concatenate them +prototypes +k +Cross-domain self-training +Domain alignment +I Semantic features Similarity paterns' I +Centers of k clusters +class prototype +( confidence +sample +1 +KL(*，*) +KL(*，*) +Cp + Source domain in support set +Clustering + Target domain in query set +Lclm +Lsfa +Lspa +Class matching loss ++ Source domain in query set +Aligment loss +Loss backpropagationcould be varied for different tasks. Meanwhile, symmetrical +alignment could bring the inference information of the +target domain to the source domain (Li et al. 2020). Thus, +we use asymmetrical KL divergence to align the distribution +of domains on both semantic features and similarity patterns +within a task. Then, KL divergence can be calculated by: +KL(A, B) =1 +2 +� +tr(Σ-1 +AΣB) + ln(ΣA +ΣB +) ++(µA − µB)Σ-1 +A(µA − µB)⊤ − d +� +, +(3) +where µA, µB, ΣA and ΣB are the mean vectors and the co- +variance matrices of sample matrix A and B, respectively. +Thus, we minimize the KL divergence between semantic +features ˆHQS and ˆHQT by +Lsfa = KL( ˆFQS, ˆFQT ). +(4) +Meanwhile, we also minimize the KL divergence to align +the similarity patterns {pc +qS} of QS and {pc +qT } of QT with +class c, which can be written by +Lspa = +N +� +c=1 +KL({pc +qS}, {pc +qT }). +(5) +In sum, we combine all the above losses, w.r.t. classifi- +cation (Eq. (1)), class matching (Eq. (2)) and KL-based do- +main alignment (Eqs. (4) and (5)) to train our model on many +episodes. The total objective function can be written by: +min Lcls + λsfaLsfa + λspaLspa + λclmLclm, +(6) +where the hyper-parameters λsfa, λspa and λclm are intro- +duced to balance the effect of different loss terms. +Experiment +DomainNet dataset. We conduct extensive experiments on a +multi-domain benchmark dataset DomainNet to demonstrate +the efficacy of our method. It was released in 2019 for the re- +search of multi-source domain adaptation (Peng et al. 2019). +It contains 345 categories and six domains per category, i.e., +quickdraw, clipart, real, sketch, painting and infograph do- +mains. In our experiments, we follow the setting of IMSE +in (Huang et al. 2021) to remove data insufficient domain +infograph. There are 20 combinations totally for evaluation, +and the dataset is split into 217, 43 and 48 categories for +episodic training, model validation and testing new tasks, +respectively. Note that in each split every category contains +the five-domain images. +Network architecture and setting. We employ ResNet- +12 as the backbone of feature embedding network, which is +widely used in few-shot learning (Huang et al. 2021) (Gi- +daris et al. 2020). We obtain semantic features by first clus- +tering the local features from each class of support set and +two query sets and then concatenating them. During this pro- +cess, we adopt cross-attention that consists of three convo- +lution parameters to generate (Q, K, V ) for attention cal- +culation. In cross-domain self-training module, we set the +threshold 1.7 of similarity score to select the ‘confidence’ +samples in target domain. The margin m in Eq. (2) is empir- +ically set to 1.5. In addition, we follow the setting of IMSE +(Huang et al. 2021) to obtain similarity patterns. The hyper- +parameters λsfa, λspa and λclm are set to 0.1, 0.05 and 0.01, +by grid search, respectively. +Model training, validation and testing. To improve the +performance, before episodic training, the feature embed- +ding network is pretrained by using source domain data in +the auxiliary dataset, as in (Huang et al. 2021). Afterwards, +we perform episodic training on 280 episodes, following the +setting of (Huang et al. 2021). During episode training, the +total loss in Eq. (6) is minimized to optimize the network +parameters for each episode. Also, we employ Adam opti- +mizer with an initial learning rate of 10-4, and meanwhile re- +duce the learning rate by half every 280 episodes. For model +validation, we compare the performance of different model +parameters on 100 tasks, which is randomly sampled from +the validate set containing 43 categories. Then, we select the +model parameters with the best validation accuracy for test- +ing. During the testing, we randomly select 3000 tasks to +calculate the averaged top-1 accuracy on these tasks as the +evaluation criterion. +Comparison Experiments for FS-UDA +We conduct extensive experiments on DomainNet to com- +pare our method with five FSL methods (ProtoNet (Snell, +Swersky, and Zemel 2017), DN4 (Li et al. 2019), ADM +(Li et al. 2020), FEAT (Ye et al. 2020), DeepEMD (Zhang +et al. 2020)), three UDA methods, (MCD (Saito et al. 2018), +ADDA (Tzeng et al. 2017), DWT (Roy et al. 2019)), their +combinations and the most related method IMSE (Huang +et al. 2021). For fair comparison, the results of these above +methods are all reported from (Huang et al. 2021) with the +same setting. Moreover, we also modify IMSE by using +our semantic features for classification and domain adver- +sary, namely IMSE+TSE. For fair comparison, these com- +pared methods also pretrain the embedding network before +episodic training, and they are trained on 1000 episodes. +Comparison analysis. Table 2 shows the results of all +the compared methods for 20 cross-domain combinations, +which records the averaged classification accuracy of tar- +get domain samples over 3000 5-way 1-shot/5-shot FS- +UDA tasks. As observed, our TSECS achieves the best per- +formance for all combinations and their average. Specifi- +cally, the UDA and FSL baselines in the first two parts per- +form the worst. In the third part, the combination methods +with ADDA (Tzeng et al. 2017) perform domain adversarial +training each episode, thus generally better than the above +two parts, but still inferior to IMSE (Huang et al. 2021) +and our TSECS. This is because the combination methods +only perform domain alignment based on original feature +maps, not considering the alignment of similarity patterns +(related to classification predictions). Also, IMSE is worse +than IMSE+TSE, which indicates high-level semantic fea- +tures are more effective for FS-UDA than local features. +However, they are still much worse than our method, show- +ing the efficacy of high-level semantic features and cross- +domain self-training for FS-UDA. +On the other hand, we can see that the 20 cross-domain +combinations have considerably different performances. +This is because several domains (e.g., quickdraw) are sig- +nificantly different from other domains, while several other +domains (e.g. real, clipart) are with the similar styles and +features. Thus, for most compared methods, the perfor- + +Table 2: Comparison of our method with the related methods for 5-way 1-shot or 5-shot FS-UDA tasks. The first three blocks +and IMSE are reported from (Huang et al. 2021), while the last two are the variant of IMSE we designed and ours, respectively. +Each row represents the accuracy (%) of a compared method adapting between two domains, where the skt, rel, qdr, pnt, and +cli denote the sketch, real, quickdraw, painting, and clipart domains in DomainNet, respectively. The best results are in bold. +5-way, 1-shot +Methods +skt ←→ rel +skt ←→ qdr +skt ←→ pnt +skt ←→ cli +rel ←→ qdr +rel ←→ pnt +rel ←→ cli +qdr ←→ pnt +qdr ←→ cli +pnt ←→ cli +avg +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +- +MCD +48.07/37.74 +38.90/34.51 +39.31/35.59 +51.43/38.98 +24.17/29.85 +43.36/47.32 +44.71/45.68 +26.14/25.02 +42.00/34.69 +39.49/37.28 +38.21 +ADDA +48.82/46.06 +38.42/40.43 +42.52/39.88 +50.67/47.16 +31.78/35.47 +43.93/45.51 +46.30/47.66 +26.57/27.46 +46.51/32.19 +39.76/41.24 +40.91 +DWT +49.43/38.67 +40.94/38.00 +44.73/39.24 +52.02/50.69 +29.82/29.99 +45.81/50.10 +52.43/51.55 +24.33/25.90 +41.47/39.56 +42.55/40.52 +41.38 +ProtoNet +50.48/43.15 +41.20/32.63 +46.33/39.69 +53.45/48.17 +32.48/25.06 +49.06/50.30 +49.98/51.95 +22.55/28.76 +36.93/40.98 +40.13/41.10 +41.21 +DN4 +52.42/47.29 +41.46/35.24 +46.64/46.55 +54.10/51.25 +33.41/27.48 +52.90/53.24 +53.84/52.84 +22.82/29.11 +36.88/43.61 +47.42/43.81 +43.61 +ADM +49.36/42.27 +40.45/30.14 +42.62/36.93 +51.34/46.64 +32.77/24.30 +45.13/51.37 +46.8/50.15 +21.43/30.12 +35.64/43.33 +41.49/40.02 +40.11 +FEAT +51.72/45.66 +40.29/35.45 +47.09/42.99 +53.69/50.59 +33.81/27.58 +52.74/53.82 +53.21/53.31 +23.10/29.39 +37.27/42.54 +44.15/44.49 +43.14 +DeepEMD +52.24/46.84 +42.12/34.77 +46.64/43.89 +55.10/49.56 +34.28/28.02 +52.73/53.26 +54.25/54.91 +22.86/28.79 +37.65/42.92 +44.11/44.38 +43.46 +ADDA+ProtoNet +51.30/43.43 +41.79/35.40 +46.02/41.40 +52.68/48.91 +37.28/27.68 +50.04/49.68 +49.83/52.58 +23.72/32.03 +38.54/44.14 +41.06/41.59 +42.45 +ADDA+DN4 +53.04/46.08 +42.64/36.46 +46.38/47.08 +54.97/51.28 +34.80/29.84 +53.09/54.05 +54.81/55.08 +23.67/31.62 +42.24/45.24 +46.25/44.40 +44.65 +ADDA+ADM +51.87/45.08 +43.91/32.38 +47.48/43.37 +54.81/51.14 +35.86/28.15 +48.88/51.61 +49.95/54.29 +23.95/33.30 +43.59/48.21 +43.52/43.83 +43.76 +ADDA+FEAT +52.72/46.08 +47.00/36.94 +47.77/45.01 +56.77/52.10 +36.32/30.50 +49.14/52.36 +52.91/53.86 +24.76/35.38 +44.66/48.82 +45.03/45.92 +45.20 +ADDA+DeepEMD +53.98/47.55 +44.64/36.19 +46.29/45.14 +55.93/50.45 +37.47/30.14 +52.21/53.32 +54.86/54.80 +23.46/32.89 +39.06/46.76 +45.39/44.65 +44.75 +IMSE +57.21/51.30 +49.71/40.91 +50.36/46.35 +59.44/54.06 +44.43/36.55 +52.98/55.06 +57.09/57.98 +30.73/38.70 +48.94/51.47 +47.42/46.52 +48.86 +IMSE+TSE +60.71/56.15 +53.78/48.57 +56.50/48.59 +61.59/56.59 +45.48/49.45 +55.44/57.45 +59.60/59.52 +37.94/39.83 +58.83/56.22 +49.19/51.01 +52.79 +TSECS (ours) +65.00/58.22 +62.25/51.97 +56.51/53.70 +69.45/64.59 +56.66/49.82 +58.76/63.18 +67.98/67.89 +38.26/46.15 +60.51/69.03 +54.40/52.76 +58.20 +5-way, 5-shot +Methods +skt ←→ rel +skt ←→ qdr +skt ←→ pnt +skt ←→ cli +rel ←→ qdr +rel ←→ pnt +rel ←→ cli +qdr ←→ pnt +qdr ←→ cli +pnt ←→ cli +avg +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +→ / ← +- +MCD +66.42/47.73 +51.84/39.73 +54.63/47.75 +72.17/53.23 +28.02/33.98 +55.74/66.43 +56.80/63.07 +28.71/29.17 +50.46/45.02 +53.99/48.24 +49.65 +ADDA +66.46/56.66 +51.37/42.33 +56.61/53.95 +69.57/65.81 +35.94/36.87 +58.11/63.56 +59.16/65.7 +723.16/33.50 +41.94/43.40 +55.21/55.86 +51.76 +DWT +67.75/54.85 +48.59/40.98 +55.40/50.64 +69.87/59.33 +36.19/36.45 +60.26/68.72 +62.92/67.28 +22.64/32.34 +47.88/50.47 +49.76/52.52 +51.74 +ProtoNet +65.07/56.21 +52.65/39.75 +55.13/52.77 +65.43/62.62 +37.77/31.01 +61.73/66.85 +63.52/66.45 +20.74/30.55 +45.49/55.86 +53.60/52.92 +51.80 +DN4 (Li et al. 2019) +63.89/51.96 +48.23/38.68 +52.57/51.62 +62.88/58.33 +37.25/29.56 +58.03/64.72 +61.10/62.25 +23.86/33.03 +41.77/49.46 +50.63/48.56 +49.41 +ADM +66.25/54.20 +53.15/35.69 +57.39/55.60 +71.73/63.42 +44.61/24.83 +59.48/69.17 +62.54/67.39 +21.13/38.83 +42.74/58.36 +56.34/52.83 +52.78 +FEAT +67.91/58.56 +52.27/40.97 +59.01/55.44 +69.37/65.95 +40.71/28.65 +63.85/71.25 +65.76/68.96 +23.73/34.02 +42.84/53.56 +57.95/54.84 +53.78 +DeepEMD +67.96/58.11 +53.34/39.70 +59.31/56.60 +70.56/64.60 +39.70/29.95 +62.99/70.93 +65.07/69.06 +23.86/34.34 +45.48/53.93 +57.60/55.61 +53.93 +ADDA+ProtoNet +66.11/58.72 +52.92/43.60 +57.23/53.90 +68.44/61.84 +45.59/38.77 +60.94/69.47 +66.30/66.10 +25.45/41.30 +46.67/56.22 +58.20/52.65 +54.52 +ADDA+DN4 +63.40/52.40 +48.37/40.12 +53.51/49.69 +64.93/58.39 +36.92/31.03 +57.08/65.92 +60.74/63.13 +25.36/34.23 +48.52/51.19 +52.16/49.62 +50.33 +ADDA+ADM +64.64/54.65 +52.56/33.42 +56.33/54.85 +70.70/63.57 +39.93/27.17 +58.63/68.70 +61.96/67.29 +21.91/39.12 +41.96/59.03 +55.57/53.39 +52.27 +ADDA+FEAT +67.80/56.71 +60.33/43.34 +57.32/58.08 +70.06/64.57 +44.13/35.62 +62.09/70.32 +57.46/67.77 +29.08/44.15 +49.62/63.38 +57.34/52.13 +55.56 +ADDA+DeepEMD +68.52/59.28 +56.78/40.03 +58.18/57.86 +70.83/65.39 +42.63/32.18 +63.82/71.54 +66.51/69.21 +26.89/42.33 +47.00/57.92 +57.81/55.23 +55.49 +IMSE +70.46/61.09 +61.57/46.86 +62.30/59.15 +76.13/67.27 +53.07/40.17 +64.41/70.63 +67.60/71.76 +33.44/48.89 +53.38/65.90 +61.28/56.74 +59.60 +IMSE+TSE +72.75/62.24 +64.49/55.04 +62.86/61.10 +77.39/69.87 +53.88/54.48 +63.97/72.46 +69.86/72.49 +37.43/51.66 +64.43/67.46 +63.40/57.89 +62.76 +TSECS (ours) +78.23/70.44 +77.90/55.77 +66.70/68.03 +83.82/74.28 +64.33/55.16 +68.40/79.74 +78.23/77.69 +39.74/63.02 +67.99/80.31 +73.67/61.63 +69.25 +Table 3: Ablation study (%) of the modules designed in +TSECS, where the FS-UDA tasks are evaluated from a do- +main (sketch) to the other four domains in DomainNet. +Components +Target Domains +TSE +catt +CS +cli +rel +qdr +pnt +✓ +61.98 +60.00 +52.21 +51.62 +✓ +57.07 +53.31 +41.93 +46.66 +✓ +✓ +62.74 +60.54 +53.64 +54.23 +✓ +✓ +68.25 +61.15 +58.31 +53.34 +✓ +✓ +✓ +69.45 +65.00 +62.25 +56.51 +mance becomes relatively low when the domain gap is large. +For example, from quickdraw to painting, it performs the +worst in all the other combinations because of larger domain +gap, but our TSECS outperforms IMSE and the other com- +pared methods by 8% and 12%, respectively. We found that +our method has the larger performance improvement over +IMSE, for these combinations containing quickdraw, which +shows the efficacy of our method for large domain gap. Also, +like TSECS, IMSE+TSE performs much better than IMSE +for large domain gap, which indicates the high-level seman- +tic features could conduct domain adaptation better than lo- +cal features. In sum, these results reflect the advantages of +our TSECS to deal with domain shift and task generaliza- +tion in FS-UDA, no matter how large the domain gap is. +Ablation study of our method. We conduct various ex- +periments on DomainNet to evaluate the effect of our mod- +ules: task-specific semantic embedding (TSE), cross-domain +self-training (CS) and cross-attention in TSE (catt). The ac- +curacies on the four target domains are reported in Table +3. As seen, our method achieve the best performance when +three modules are all used. The performance of the single +CS is the worst that shows that local features cannot align +the distributions of the two domains, thus affecting cross- +domain self-training. The module TSE is introduced into +four combinations, all improving the performance, which +validates the efficacy of our task-specific semantic features +for FS-UDA again. Also, the addition of cross-attention into +TSE will further improve the performance, which can help +discover more semantics from previous tasks. +Ablation study of different losses. We conduct various +experiments on DomainNet to further evaluate the effect of +different losses in Eq. (6). Besides the classification loss +(Lcls), we combine the remaining three loss terms: 1) se- +mantic features alignment loss (Lsfa), 2) similarity pattern +alignment loss (Lspa), and 3) class matching loss (Lclm). +We evaluate 5-way 1-shot FS-UDA tasks from sketch to the +other four domains, respectively, and their accuracies are re- +ported in Table 4. As observed, the more the number of loss +terms involved, the higher the accuracy. The combination of +all the three losses is the best. For the single loss, both Lsfa + +Table 4: Ablation study (%) of the three losses designed in +TSECS, where the FS-UDA tasks are evaluated from a do- +main (sketch) to the other four domains in DomainNet. +Components +Target Domains +Lsfa +Lspa +Lclm +cli +rel +qdr +pnt +✓ +66.67 +58.84 +56.91 +43.28 +✓ +64.28 +57.32 +52.11 +42.46 +✓ +66.83 +58.29 +56.51 +44.25 +✓ +✓ +66.64 +62.64 +57.41 +53.40 +✓ +✓ +68.04 +63.98 +59.13 +55.39 +✓ +✓ +67.61 +62.47 +53.07 +54.14 +✓ +✓ +✓ +69.45 +65.00 +62.25 +56.51 +Figure 4: Comparison of introducing our TSE module or not +into two FSL methods with ADDA (Tzeng et al. 2017) com- +bined, i.e., ADDA+ProtoNet and ADDA+DN4. +and Lclm perform better than Lspa, and their combination is +also considerably better than the other paired combinations, +showing the efficacy of semantic feature domain alignment +and class matching in target domain. Based on the above, +adding Lspa further improves the performance, indicating +positive effect of aligning the similarity patterns. +Evaluation on the effect of our task-specific se- +mantic embedding module on two FSL methods with +ADDA (Tzeng et al. 2017) combined. Compared with +ADDA+DN4 and ADDA+ProtoNet, we add our semantic +embedding module (TSE) with the loss Lsfa into their fea- +ture embedding models, and test them on 3000 new 5-way +1/5-shot FS-UDA tasks. For simplification and clarification, +we calculate the averaged accuracies from every domain to +the other four domains and show them in Fig. 4. As seen, +the methods using TSE generally perform better than that +without it, which validates that the semantic embedding in +TSE could generate more discriminative semantic features +for classification than original local features. In addition, the +performances of these methods are still far from our method +because using ADDA is insufficient to align the domains and +could result in class mismatching, but our method can effec- +tively solve it by cross-domain self-training. +Evaluation of dataset generalization. We evaluate the +generalization of our model trained on DomainNet to adapt +to a substantially different dataset miniImageNet. We mod- +ify miniImageNet by transferring a half of real images (rel) +into sketch images (skt) by MUNIT (Huang et al. 2018) to +Table 5: Evaluation (%) of dataset generalization for 5-way +1-shot FS-UDA tasks between domains real and sketch, per- +forming episodic training on DomainNet and testing on ex- +panded dataset miniImageNet. +Methods +skt → rel +rel → skt +ADDA+DN4 +44.01 ± 0.87 +40.61 ± 0.90 +ADDA+DeepEMD +46.14 ± 0.82 +45.91 ± 0.77 +IMSE +48.78 ± 0.78 +48.52 ± 0.81 +TSECS (ours) +53.33 ± 1.08 +49.83 ± 0.96 +Figure 5: The tSNE visualization of our TSECS using cross- +domain self-training or not for a 5-way 5-shot FS-UDA task +from sketch to clipart. The samples with different colors be- +long to different classes, and the stars in the left and right +figures represent the class centroids of support set and se- +lected target domain query samples, respectively. +produce two domains for FS-UDA. We compare our method +with ADDA+DN4, ADDA+DeepEMD and IMSE for 5-way +1-shot FS-UDA tasks for rel ↔ skt. The results are shown +as Table 5. As observed, our method outperforms other +methods, specially for ske → rel. For rel → skt, our method +is slightly better than IMSE, because the style of sketch im- +ages in miniImageNet is relatively different from that in Do- +mainNet, which could effect the learned semantic features. +Visualization of our method using cross-domain self- +training or not. We illustrate the tSNE results of a 5-way 5- +shot FS-UDA task from sketch to clipart in Fig. 5. Note that +the class prototypes in the left subfigure belong to the sup- +port set in source domain, while those in the right subfigure +are generated by ‘confidence’ samples in target domain. It +is obvious that two class prototypes in the left subfigure are +fully overlapped so that many samples could not be correctly +classified. In contrast, the right subfigure has the better class +prototypes, and samples from different classes are more dis- +tinguishable. This shows the efficacy of our cross-domain +self-training that finds ‘confidence’ samples to train the tar- +get domain classifier and uses class matching loss Lclm to +shorten the distance of samples of the same class. +Conclusion +In this paper, we propose a novel method TSECS for FS- +UDA. We extract high-level semantic features than local fea- +tures to measure the similarity of query images in target do- +main to support classes in source domain. Moreover, we de- +sign cross-domain self-training to train a target domain clas- +sifier. In addition, asymmetrical KL-divergence is used to +align the semantic features between domains. Extensive ex- +periments on DomainNet show the efficacy of our TSECS, +significantly improving the performance for FS-UDA. + +ProtoNet/1-shot +.- +DN4/1-shot +701 +701 +60 +60 +50 +50 +40 +40 +30 +30 +skt +cli rel qdr +pnt +skt +clirel qdr +pnt +ProtoNet/5-shot +DN4/5-shot +70 +70 +60 +60 +50 +50 +40 +40 +30 +30 +skt +clirel +Ipb +pnt +skt +clirel qdr +pnt +ADDA+ProtoNet +ADDA+DN4 +ADDA+TSE+ProtoNet +ADDA+TSE+DN4 +ours +oursTSECS (no CS) +TSECS +1.0 +1.0 +0.8 +0.8 +0.6 +0.6 +0.4 +0.4 +. +★ +0.2 +. +0.2 +. +： +* +. +0.0 +. +0.0 +. +. +. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0Acknowledgments +Wanqi Yang and Ming Yang are supported by Na- +tional Natural Science Foundation of China (Grant Nos. +62076135, 62276138, 61876087). Lei Wang is supported +by an Australian Research Council Discovery Project (No. +DP200101289) funded by the Australian Government. +References +Bertinetto, L.; Henriques, J. F.; Torr, P.; and Vedaldi, +A. 2019. +Meta-learning with differentiable closed-form +solvers. In International Conference on Learning Represen- +tations, 1–8. +Chen, W.-Y.; Liu, Y.-C.; Kira, Z.; Wang, Y.-C. F.; and +Huang, J.-B. 2019. A Closer Look at Few-shot Classifica- +tion. In International Conference on Learning Representa- +tions, 1–16. +Finn, C.; Abbeel, P.; and Levine, S. 2017. Model-Agnostic +Meta-Learning for Fast Adaptation of Deep Networks. In +Precup, D.; and Teh, Y. W., eds., Proceedings of the 34th +International Conference on Machine Learning, volume 70 +of Proceedings of Machine Learning Research, 1126–1135. +Ganin, Y.; Ustinova, E.; Ajakan, H.; Germain, P.; Larochelle, +H.; Laviolette, F.; March, M.; and Lempitsky, V. 2016. +Domain-Adversarial Training of Neural Networks. Journal +of Machine Learning Research, 17(59): 1–35. +Gidaris, S.; Bursuc, A.; Komodakis, N.; Perez, P.; and Cord, +M. 2020. Learning Representations by Predicting Bags of +Visual Words. +In Proceedings of the IEEE/CVF Confer- +ence on Computer Vision and Pattern Recognition (CVPR), +6926–6936. +Huang, S.; Yang, W.; Wang, L.; Zhou, L.; and Yang, M. +2021. +Few-Shot Unsupervised Domain Adaptation with +Image-to-Class Sparse Similarity Encoding. In Proceedings +of the 29th ACM International Conference on Multimedia, +MM ’21, 677–685. New York, NY, USA: Association for +Computing Machinery. ISBN 9781450386517. +Huang, X.; Liu, M.-Y.; Belongie, S.; and Kautz, J. 2018. +Multimodal Unsupervised Image-to-Image Translation. In +Ferrari, V.; Hebert, M.; Sminchisescu, C.; and Weiss, Y., +eds., Computer Vision – ECCV 2018, 179–196. Cham: +Springer International Publishing. ISBN 978-3-030-01219- +9. +Kim, D.; Saito, K.; Oh, T.-H.; Plummer, B. A.; Sclaroff, S.; +and Saenko, K. 2021. CDS: Cross-Domain Self-supervised +Pre-training. In 2021 IEEE/CVF International Conference +on Computer Vision (ICCV), 9103–9112. +Li, W.; Wang, L.; Huo, J.; Shi, Y.; Gao, Y.; and Luo, J. 2020. +Asymmetric Distribution Measure for Few-shot Learning. In +Bessiere, C., ed., Proceedings of the Twenty-Ninth Interna- +tional Joint Conference on Artificial Intelligence, IJCAI-20, +2957–2963. International Joint Conferences on Artificial In- +telligence Organization. Main track. +Li, W.; Wang, L.; Xu, J.; Huo, J.; Gao, Y.; and Luo, J. 2019. +Revisiting Local Descriptor Based Image-To-Class Measure +for Few-Shot Learning. In Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recognition +(CVPR), 7260–7268. +Long, M.; Cao, Y.; Wang, J.; and Jordan, M. 2015. Learn- +ing Transferable Features with Deep Adaptation Networks. +In Bach, F.; and Blei, D., eds., Proceedings of the 32nd In- +ternational Conference on Machine Learning, volume 37 of +Proceedings of Machine Learning Research, 97–105. Lille, +France: PMLR. +Luo, Y.; Liu, P.; Guan, T.; Yu, J.; and Yang, Y. 2020. Ad- +versarial Style Mining for One-Shot Unsupervised Domain +Adaptation. +In Larochelle, H.; Ranzato, M.; Hadsell, R.; +Balcan, M.; and Lin, H., eds., Advances in Neural Informa- +tion Processing Systems, volume 33, 20612–20623. Curran +Associates, Inc. +Peng, X.; Bai, Q.; Xia, X.; Huang, Z.; Saenko, K.; and +Wang, B. 2019. Moment Matching for Multi-Source Do- +main Adaptation. In 2019 IEEE/CVF International Confer- +ence on Computer Vision (ICCV), 1406–1415. +Ravi, S.; and Larochelle, H. 2017. Optimization as a Model +for Few-Shot Learning. +In International Conference on +Learning Representations. +Roy, S.; Siarohin, A.; Sangineto, E.; Bulo, S. R.; Sebe, N.; +and Ricci, E. 2019. Unsupervised Domain Adaptation Using +Feature-Whitening and Consensus Loss. In Proceedings of +the IEEE/CVF Conference on Computer Vision and Pattern +Recognition (CVPR), 9471–9480. +Saito, K.; Watanabe, K.; Ushiku, Y.; and Harada, T. 2018. +Maximum Classifier Discrepancy for Unsupervised Domain +Adaptation. +In Proceedings of the IEEE Conference on +Computer Vision and Pattern Recognition (CVPR), 3723– +3732. +Snell, J.; Swersky, K.; and Zemel, R. 2017. Prototypical Net- +works for Few-shot Learning. In Guyon, I.; Luxburg, U. V.; +Bengio, S.; Wallach, H.; Fergus, R.; Vishwanathan, S.; and +Garnett, R., eds., Advances in Neural Information Process- +ing Systems, volume 30, 4077–4087. Curran Associates, Inc. +Tang, H.; Chen, K.; and Jia, K. 2020. Unsupervised Domain +Adaptation via Structurally Regularized Deep Clustering. In +Proceedings of the IEEE/CVF Conference on Computer Vi- +sion and Pattern Recognition (CVPR). +Tseng, H.-Y.; Lee, H.-Y.; Huang, J.-B.; and Yang, M.-H. +2020. Cross-Domain Few-Shot Classification via Learned +Feature-Wise Transformation. In International Conference +on Learning Representations. +Tzeng, E.; Hoffman, J.; Saenko, K.; and Darrell, T. 2017. +Adversarial Discriminative Domain Adaptation. +In 2017 +IEEE Conference on Computer Vision and Pattern Recog- +nition (CVPR), 2962–2971. +Tzeng, E.; Hoffman, J.; Zhang, N.; Saenko, K.; and Darrell, +T. 2014. Deep Domain Confusion: Maximizing for Domain +Invariance. CoRR, abs/1412.3474: 1–9. +Vinyals, O.; Blundell, C.; Lillicrap, T.; kavukcuoglu, k.; and +Wierstra, D. 2016. Matching Networks for One Shot Learn- +ing. In Lee, D.; Sugiyama, M.; Luxburg, U.; Guyon, I.; and +Garnett, R., eds., Advances in Neural Information Process- +ing Systems, volume 29, 3630–3638. Curran Associates, Inc. +Yang, W.; Yang, C.; Huang, S.; Wang, L.; and Yang, M. +2022. Few-shot Unsupervised Domain Adaptation via Meta + +Learning. In IEEE International Conference on Multimedia +and Expo (ICME). +Ye, H.-J.; Hu, H.; Zhan, D.-C.; and Sha, F. 2020. Few-Shot +Learning via Embedding Adaptation With Set-to-Set Func- +tions. In Proceedings of the IEEE/CVF Conference on Com- +puter Vision and Pattern Recognition (CVPR), 8805–8814. +Yue, X.; Zheng, Z.; Zhang, S.; Gao, Y.; Darrell, T.; Keutzer, +K.; and Vincentelli, A. S. 2021. Prototypical Cross-Domain +Self-Supervised Learning for Few-Shot Unsupervised Do- +main Adaptation. In Proceedings of the IEEE/CVF Confer- +ence on Computer Vision and Pattern Recognition (CVPR), +13834–13844. +Zhang, C.; Cai, Y.; Lin, G.; and Shen, C. 2020. +Deep- +EMD: Few-Shot Image Classification With Differentiable +Earth Mover’s Distance and Structured Classifiers. In Pro- +ceedings of the IEEE/CVF Conference on Computer Vision +and Pattern Recognition (CVPR), 12200–12210. +Zou, Y.; Yu, Z.; Liu, X.; Kumar, B. V. K. V.; and Wang, +J. 2019. +Confidence Regularized Self-Training. +In 2019 +IEEE/CVF International Conference on Computer Vision +(ICCV), 5981–5990. +Zou, Y.; Yu, Z.; Vijaya Kumar, B. V. K.; and Wang, J. 2018. +Unsupervised Domain Adaptation for Semantic Segmenta- +tion via Class-Balanced Self-training. In Ferrari, V.; Hebert, +M.; Sminchisescu, C.; and Weiss, Y., eds., Computer Vision +– ECCV 2018, 297–313. Cham: Springer International Pub- +lishing. ISBN 978-3-030-01219-9. + diff --git a/A9AzT4oBgHgl3EQf__9t/content/tmp_files/load_file.txt b/A9AzT4oBgHgl3EQf__9t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..87001685b86d1119d1c4a172059a70f4523c0048 --- /dev/null +++ b/A9AzT4oBgHgl3EQf__9t/content/tmp_files/load_file.txt @@ -0,0 +1,1616 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf,len=1615 +page_content='High-level semantic feature matters few-shot unsupervised domain adaptation Lei Yu1, Wanqi Yang1*, Shengqi Huang1, Lei Wang2, Ming Yang1 1School of Computer and Electronic Information, Nanjing Normal University, China 2School of Computing and Information Technology, University of Wollongong, Australia yulei@njnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='cn, yangwq@njnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='cn, huangshengqi@njnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='cn, leiw@uow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='au, myang@njnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='cn Abstract In few-shot unsupervised domain adaptation (FS-UDA), most existing methods followed the few-shot learning (FSL) meth- ods to leverage the low-level local features (learned from con- ventional convolutional models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', ResNet) for classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' However, the goal of FS-UDA and FSL are relevant yet distinct, since FS-UDA aims to classify the samples in target domain rather than source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We found that the local features are insufficient to FS-UDA, which could introduce noise or bias against classification, and not be used to effec- tively align the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' To address the above issues, we aim to refine the local features to be more discriminative and rele- vant to classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Thus, we propose a novel task-specific semantic feature learning method (TSECS) for FS-UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' TSECS learns high-level semantic features for image-to-class similarity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Based on the high-level features, we design a cross-domain self-training strategy to leverage the few labeled samples in source domain to build the classi- fier in target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In addition, we minimize the KL diver- gence of the high-level feature distributions between source and target domains to shorten the distance of the samples be- tween the two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Extensive experiments on Domain- Net show that the proposed method significantly outperforms SOTA methods in FS-UDA by a large margin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', ∼ 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' keywords Few-shot unsupervised domain adaptation, image-to-class similarity, high-level semantic features, cross-domain self- training, cross-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Introduction Currently, a setting namely few-shot unsupervised domain adaptation (FS-UDA) (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021)(Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2022), which utilizes few labeled data in source domain to train a model to classify unlabeled data in target domain, owns its potential feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Typically, a FS-UDA model could learn general knowledge from base classes during training to guide classification in novel classes during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' It is known that both insufficient labels in source domain and large domain shift make FS-UDA as a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Previous studies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', IMSE (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021), first fol- lowed several few-shot learning (FSL) methods (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The corresponding author is Wanqi Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Figure 1: A 5-way 1-shot task for FS-UDA where the sup- port set includes five classes and one sample for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The figure shows the similarity of query images to every support classes and the spatial similarity of query images to the predicted support class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We found using local fea- tures could cause some inaccurate regions of query images to match the incorrect classes, while our semantic features make the object region in query images similar with their true class, thus achieving correct classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019)(Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017) to learn the local features by us- ing convolutional models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', ResNet) and then leveraged them to learn image-to-class similarity pattern for classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' However, we wish to clarify that the goal of FS-UDA and FSL are relevant yet distinct, since both of them suf- fer from insufficient labeled training data whereas FS-UDA aims to classify the samples in target domain rather than source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 1, by visualizing the spatial similarity of query images to predicted support classes, we found using local features causes the inaccurate regions of query images to match incorrect classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' This reason might be that few labeled samples and large domain shift between the support and query sets simultaneously result in the con- ventional local features in FSL to fail in classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In this sense, the local features are insufficient to FS-UDA, which could introduce noise or bias against the classification in tar- get domain and not be used to effectively align the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' To address this issue, we aim to refine the low-level local arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='01956v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='CV] 5 Jan 2023 support set in the source domain (sketch) sailboat bed glasses television snowman query set in the target domain (clipart) local features semantic features (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='4 0.' 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+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='1 television as saFigure 2: Illustration of the process for cross-domain self- training in TSECS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Different shapes represent different do- mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We first select the ‘confidence’ target samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', a) that are very similar to support classes, and then regard them as the new class prototypes to further classify the other target samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' This process is executed itera- tively with using class matching loss to narrow the distance of query images and their most similar support classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' features to be more discriminative and relevant to classifica- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', high-level semantic features, and meanwhile align the semantic features for domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Therefore, we propose a novel task-specific semantic feature method (TSECS) that learns the semantic features for each task by clustering the local features of support set and query set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' To obtain the related semantics from previous tasks, the cluster centroids of the current task are then fused by cross-attention with that of the previous task to generate high-level semantic features to boost classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Moreover, for the domain shift between source and tar- get domains, many domain adaptation methods (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2018)(Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017)(Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2014) reduced the dis- tribution discrepancy between domains by using a discrim- inator to adverse against feature embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' However, this way could fail in aligning the samples of the same class be- tween domains due to label missing in target domain, which could make the classes of two domains mismatched and thus affect the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Therefore, we aim to align the high- level semantic features by minimizing the KL divergence of the semantic feature distributions between domains, and meanwhile design a cross-domain self-training strategy to train the classifier in target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We hypothesis that there are usually several ‘confidence’ samples in target domain that could be classified correctly by support set in source domain, in other words, they are very similar to their class prototypes in source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Mean- while, the target domain samples in the same class are more similar to each other than that of other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Based on this, we regard these ‘confidence’ samples in the target domain as new prototypes of the classes, which replace those from the support set of source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2, several ‘confidence’ samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', a) can be selected as prototypes of their similar classes for classification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', b and c) in tar- get domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Moreover, the process is conducted iteratively by using class matching loss for better domain alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In sum, we propose the novel method, namely TSECS, for FS-UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' It refines the local features of convolutional network to generate specific semantic features of each task, and meanwhile perform cross-domain self-training to trans- port labels from support set in the source domain to query set in the target domain to effectively classify the samples in target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Our contributions can be summarized as: (1) A novel solution for FS-UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' TSECS aims to learn high-level semantic features for classification and do- main alignment, which could be regarded as a more ef- fective and efficient way than using local features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (2) Task-specific semantic embedding for few-shot set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' It can be seamlessly add to existing FSL/FS-UDA models, which could alleviate the bias of classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (3) Cross-domain self-training for domain alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' It is designed to bring the samples of the same class close, which could guide effective domain alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We conduct extensive experiments on DomainNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Our method significantly outperforms SOTA methods in FS- UDA by a large margin up to ∼ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Related Works Unsupervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The conventional UDA methods aim to reduce discrepancy between source domain and target domain in the feature space and utilize suffi- ciently labeled source domain data to classify data from tar- get domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The difference between unsupervised domain adaptation methods often lies in the evaluation of domain discrepancy and the objective function of model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Several researchers (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2015)(Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2014) minimize the feature discrepancy by using maximum mean discrepancy to measure the discrepancy between the distri- bution of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Moreover, adversarial training (Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017)(Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2016) to learn domain-invariant fea- tures is usually used to tackle domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Several meth- ods (Tang, Chen, and Jia 2020)(Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019)(Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2018)(Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021)train the classifier in both source do- main and target domain and utilize pseudo-labels from target domain to calculate classification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Overall, these UDA methods all require sufficiently labeled source domain data to realize domain alignment and classification, but they per- form poor when labeled source domain data are insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Few-shot learning has two main streams, metric-based and optimization-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Optimization-based methods (Bertinetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019)(Finn, Abbeel, and Levine 2017)(Ravi and Larochelle 2017) usu- ally train a meta learner over auxiliary dataset to learn a general initialization model, which can fine-tune and adapt to new tasks very soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The main purpose of metric- based methods (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019)(Snell, Swersky, and Zemel 2017)(Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2016)(Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020) is that learn a gen- eralizable feature embedding for metric learning, which can immediately adapt to new tasks without any fine-tune and retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Typically, ProtoNet (Snell, Swersky, and Zemel 2017) learns the class prototypes in the support set and clas- sifies the query images based on the maximum similarity to these prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Other than these metric-based methods on feature maps, many methods on local features have ap- peared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' DN4 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019) utilizes large amount of local features to measure the similarity between support and query select \'confidence" sanples use new prototypes for as new prototypes classification in target domain O b O 0 00 00 lass natching loss 0 00 dims prototypes doeifiad query imega [sonmce dm ngin) (trt domain) at din) (trt domain)sets instead of flattening the feature map into a long vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Based on local features, DeepEMD (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020) adopts Earth Mover’s Distance distance to measure the re- lationship between query and support sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Furthermore, a few recent works focus on the issue of cross-domain FSL in which domain shift exists between data of meta tasks and new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The baseline models (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019) are used to do cross-domain FSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' LFT (Tseng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020) performs adaptive feature transformation to tackle the domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Few-shot unsupervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Compared with UDA, FS-UDA is to deal with many UDA tasks by leveraging few labeled source domain samples for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' And compared with cross-domain FSL, FS-UDA are capable of handling the circumstances of no available labels in the tar- get domain, and large domain gap between the support and query sets in every task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' For the one-shot UDA (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020), it deals with the case that only one unlabeled target sample is available, but does not require the source domain to be few-shot, which is different from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Recently, there are a few attempts in FS-UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' PCS (Yue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021) per- forms prototype self-supervised learning in cross-domain, but they require enough unlabeled source samples to learn prototypes and ignore task-level transfer, which is also dif- ferent from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' meta-FUDA (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2022) lever- ages meta learning-based optimization to perform task-level transfer and domain-level transfer jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' IMSE (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021) utilizes local features to learn similarity patterns for cross-domain similarity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' However, they did not consider that local features could bring the noise or bias to affect classification and domain alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Thus, we pro- pose task-specific semantic features to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Methodology Problem Definition A N-way, K-shot FS-UDA task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Table 1 shows the main symbols used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The FS-UDA setting includes two domains: a source domain S and a target domain T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' A N-way, K-shot FS-UDA task includes a support set XS from S and a query set QT from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The support set XS contains N classes and K samples per class in the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The query set QT contains the same N classes as in XS and Nq target domain samples per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' To classify query images in QT to the correct class in XS, it is popular to train a general model from base classes to adapt to handle new N-way, K-shot FS-UDA tasks for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Auxiliary dataset and episodic training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' As in (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021), the base classes are collected from an auxil- iary dataset Daux to perform episodic training to learn the general model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Note that the base classes in Daux are com- pletely different from new classes in testing tasks, which are unseen during episodic training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Moreover, Daux includes labeled source domain data and unlabeled target domain data for FS-UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We construct large amounts of episodes, each containing {XS, QS, QT } as in (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021), to simulate the testing tasks for task-level generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Note that QS is introduced into episodic training to calculate clas- sification loss and perform domain alignment with QT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The flowchart of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 3 illustrates our Table 1: Notations Notations Descriptions N ∈ R The number of classes in the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' K ∈ R The number of samples per class in support set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' XS, QS, QT Support set of source domain, and query sets of source domain and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' H, W, d ∈ R The height, width, and channel of feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' L ∈ RHW ×d The local feature vectors in the feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' k ∈ R The number of semantic clusters for an episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' C ∈ Rk×d The centroids of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' F, ˆF, The semantic feature map, semantic features and ˆFXS, ˆFQS, ˆFQT the parts of support and query sets in both domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' M c q ∈ RH×W ×N The 3-D similarity matrix for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' pc q ∈ RKHW Similarity pattern vectors of a query image q pi q ∈ RHW with a support class c and a support image i, ppos q , pneg q ∈ RKHW and the most similar class and the second one for q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' µA, µB ∈ RHW ×d The mean of semantic features or similarity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ΣA, ΣB ∈ RHW ×HW Covariance matrix of semantic features or similarity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' λsfa, λspa, λclm Weight parameters of three loss terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' method for 5-way, 1-shot FS-UDA tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In each episode, a support set (XS) and two query sets (QS and QT ) are first through the convolution network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', ResNet) to extract their local features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Then, the task-specific semantic embed- ding module refines the local features to generate semantic features, which is computational efficient due to dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Also, based on semantic features of QS and QT , we leverage their similarity patterns (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021) to calculate image-to-class similarity for classification with the loss Lcls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' To improve its performance, cross-domain self- training module is performed to introduce the class proto- types of target domain and train a target domain classifier with a class matching loss Lclm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In addition, the seman- tic features and similarity patterns from both domains are further aligned by calculating their alignment losses Lsfa and Lspa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Finally, the losses above are back- propagated to update our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' After episodic training over all episodes, we utilize the learned model to test new FS- UDA tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Then, we calculate the averaged classification accuracy on these tasks for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Task-specific Semantic Feature Learning Most FSL methods and FS-UDA methods learned local fea- tures from convolutional networks for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' How- ever, we found that the local features could introduce noise or bias that is valid for classification and domain alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Thus, we aim to refine the local features to generate high- level semantic features for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In the following, we will introduce our semantic feature embedding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' First of all, in each episode, all local features L ∈ R(|XS|+|QS|+|QT |)HW ×d are extracted from the convolu- tional network, where | · | is the number of samples in a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Then, we cluster the local features to generate different semantic clusters for support set and query set, respectively, since clustering the two sets together could result in the clus- ters that relate to the domains due to the presence of large do- main gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' For simplification, we adopt K-means for cluster- ing, and meanwhile utilize the singular value decomposition (SVD) to adaptively take the number of eigenvalues greater than a certain threshold as the cluster number k (k ≪ d) for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Afterwards, we calculate the task-specific seman- Figure 3: Illustration of our method training per episode for 1-shot FS-UDA tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' First, support classes and query images from both domains are through a convolution network to extract their local features, followed by the task-specific semantic embedding module to learn high-level semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Then, these semantic features are fed into the cross-domain self- training module to update the class prototypes for target domain classification and calculate the class matching loss Lclm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Meanwhile, these semantic features are also used to generate similarity patterns in IMSE (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021) for classification loss Lcls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In addition, both semantic features and similarity patterns from both domains are aligned by the domain alignment module with the alignment losses Lsfa and Lspa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Finally, all the losses are backpropagated to update our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' tic feature map F ∈ R(|XS|+|QS|+|QT |)HW ×k by measuring the Cosine similarity between the local features L and the centroids C ∈ Rk×d of all semantic clusters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', F = L ||L||2 · C⊤ ||C||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Finally, we split F to 2×2 blocks based on height and weight dimension of the feature map, and then concatenate the four blocks together along the channel to generate se- mantic features ˆF ∈ R 1 4 (|XS|+|QS|+|QT |)HW ×4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' This is a simple yet effective way to maintain discriminative ability and spatial information of semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Moreover, to leverage the semantics from previous tasks to guide the semantic feature learning of the current task, we utilize the centroids of previous clusters to update the initial- ization of clustering centroids by cross-attention (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' This makes K-means clustering converge rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' After obtaining the semantic features ˆF, we use them for domain alignment and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Firstly, ˆF is partitioned into ˆFXS, ˆFQS, ˆFQT along with the first dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Then, we align ˆFQS and ˆFQT by minimizing the KL divergence of their distributions that will be introduced later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Meanwhile, we utilize ˆFXS, ˆFQS and ˆFQT to build 3-D similarity matrix M c q (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021) between support and query sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Fi- nally, we calculate the similarity pattern pc q (measuring the similarity between query sample q and support class c) for classification (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The classification loss us- ing cross-entropy can be written by: Lcls = − 1 |QS| � q∈QS log( exp(1 · pc q) �K i=1 exp(1 · piq) ) (1) Cross-domain Self-training Since there is large domain shift between source and target domains, as well as label missing in target domain, adver- sarial domain adaptation on low-level local features cannot make samples of the same class between domains close, and thus could make the classes of two domains mismatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' To alleviate the mismatching issue, we aim to find the most similar ‘confidence’ samples in QT with XS to guide classification in target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We assume that it usually exists that the ‘confidence’ samples in QT could be clas- sified correctly by XS, when the distributions between do- mains are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We iteratively select the ‘confidence’ samples in QT as the new prototypes to replace that in XS for classification, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We call the process as cross-domain self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The process can find more ‘con- fidence’ samples from QT than that in XS for the same class, which could correct some misclassified samples in QT , thereby lightening the impact of domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Moreover, to improve the performance of the target do- main classifier, we aim to make target domain samples q in QT closer to their most similar class and meanwhile far away from the other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Thus, we first calculate its sim- ilarity patterns ppos q (with the most similar class) and pneg q (with the second similar class), and then design the class matching loss with a margin m, which can be written by Lclm = � q∈QT max(softmax(pneg q )−softmax(ppos q )+m, 0), (2) where the similarity to the most similar class should be greater by m than the second similar class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Two-level Domain Alignment Conventional adversarial domain adaptation methods (Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2016)(Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017) iteratively train a discriminator to align the distribution of domains by adver- sarial training among tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' they cannot be used to align the semantic features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' because our semantic features are relevant to tasks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' the semantics of the same channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Task-specific semantic embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Local features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Semantic feature maps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='High-level semantic features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Support class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='(Source domain) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Similarity patterns in IMSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Qurey image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='MH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='(Target domain) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Lcls ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Classification loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Query image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='(Source domain) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Split into 2 x 2 blocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='I Update the class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='and concatenate them ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='prototypes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Cross-domain self-training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Domain alignment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='I Semantic features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content="Similarity paterns' I " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='Centers of k clusters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='class prototype ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='( confidence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='sample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='KL(*，' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='*) KL(*，' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='*) Cp Source domain in support set Clustering Target domain in query set Lclm Lsfa Lspa Class matching loss + Source domain in query set Aligment loss Loss backpropagationcould be varied for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Meanwhile, symmetrical alignment could bring the inference information of the target domain to the source domain (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Thus, we use asymmetrical KL divergence to align the distribution of domains on both semantic features and similarity patterns within a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Then, KL divergence can be calculated by: KL(A, B) =1 2 � tr(Σ-1 AΣB) + ln(ΣA ΣB ) +(µA − µB)Σ-1 A(µA − µB)⊤ − d � , (3) where µA, µB, ΣA and ΣB are the mean vectors and the co- variance matrices of sample matrix A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Thus, we minimize the KL divergence between semantic features ˆHQS and ˆHQT by Lsfa = KL( ˆFQS, ˆFQT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (4) Meanwhile, we also minimize the KL divergence to align the similarity patterns {pc qS} of QS and {pc qT } of QT with class c, which can be written by Lspa = N � c=1 KL({pc qS}, {pc qT }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (5) In sum, we combine all the above losses, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' classifi- cation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (1)), class matching (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (2)) and KL-based do- main alignment (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (4) and (5)) to train our model on many episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The total objective function can be written by: min Lcls + λsfaLsfa + λspaLspa + λclmLclm, (6) where the hyper-parameters λsfa, λspa and λclm are intro- duced to balance the effect of different loss terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Experiment DomainNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We conduct extensive experiments on a multi-domain benchmark dataset DomainNet to demonstrate the efficacy of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' It was released in 2019 for the re- search of multi-source domain adaptation (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' It contains 345 categories and six domains per category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', quickdraw, clipart, real, sketch, painting and infograph do- mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In our experiments, we follow the setting of IMSE in (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021) to remove data insufficient domain infograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' There are 20 combinations totally for evaluation, and the dataset is split into 217, 43 and 48 categories for episodic training, model validation and testing new tasks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Note that in each split every category contains the five-domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Network architecture and setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We employ ResNet- 12 as the backbone of feature embedding network, which is widely used in few-shot learning (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021) (Gi- daris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We obtain semantic features by first clus- tering the local features from each class of support set and two query sets and then concatenating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' During this pro- cess, we adopt cross-attention that consists of three convo- lution parameters to generate (Q, K, V ) for attention cal- culation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In cross-domain self-training module, we set the threshold 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='7 of similarity score to select the ‘confidence’ samples in target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The margin m in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (2) is empir- ically set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In addition, we follow the setting of IMSE (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021) to obtain similarity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The hyper- parameters λsfa, λspa and λclm are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='01, by grid search, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Model training, validation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' To improve the performance, before episodic training, the feature embed- ding network is pretrained by using source domain data in the auxiliary dataset, as in (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Afterwards, we perform episodic training on 280 episodes, following the setting of (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' During episode training, the total loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (6) is minimized to optimize the network parameters for each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Also, we employ Adam opti- mizer with an initial learning rate of 10-4, and meanwhile re- duce the learning rate by half every 280 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' For model validation, we compare the performance of different model parameters on 100 tasks, which is randomly sampled from the validate set containing 43 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Then, we select the model parameters with the best validation accuracy for test- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' During the testing, we randomly select 3000 tasks to calculate the averaged top-1 accuracy on these tasks as the evaluation criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Comparison Experiments for FS-UDA We conduct extensive experiments on DomainNet to com- pare our method with five FSL methods (ProtoNet (Snell, Swersky, and Zemel 2017), DN4 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019), ADM (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020), FEAT (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020), DeepEMD (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020)), three UDA methods, (MCD (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2018), ADDA (Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017), DWT (Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019)), their combinations and the most related method IMSE (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' For fair comparison, the results of these above methods are all reported from (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021) with the same setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Moreover, we also modify IMSE by using our semantic features for classification and domain adver- sary, namely IMSE+TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' For fair comparison, these com- pared methods also pretrain the embedding network before episodic training, and they are trained on 1000 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Comparison analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Table 2 shows the results of all the compared methods for 20 cross-domain combinations, which records the averaged classification accuracy of tar- get domain samples over 3000 5-way 1-shot/5-shot FS- UDA tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' As observed, our TSECS achieves the best per- formance for all combinations and their average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Specifi- cally, the UDA and FSL baselines in the first two parts per- form the worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In the third part, the combination methods with ADDA (Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017) perform domain adversarial training each episode, thus generally better than the above two parts, but still inferior to IMSE (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021) and our TSECS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' This is because the combination methods only perform domain alignment based on original feature maps, not considering the alignment of similarity patterns (related to classification predictions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Also, IMSE is worse than IMSE+TSE, which indicates high-level semantic fea- tures are more effective for FS-UDA than local features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' However, they are still much worse than our method, show- ing the efficacy of high-level semantic features and cross- domain self-training for FS-UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' On the other hand, we can see that the 20 cross-domain combinations have considerably different performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' This is because several domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', quickdraw) are sig- nificantly different from other domains, while several other domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' real, clipart) are with the similar styles and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Thus, for most compared methods, the perfor- Table 2: Comparison of our method with the related methods for 5-way 1-shot or 5-shot FS-UDA tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The first three blocks and IMSE are reported from (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021), while the last two are the variant of IMSE we designed and ours, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Each row represents the accuracy (%) of a compared method adapting between two domains, where the skt, rel, qdr, pnt, and cli denote the sketch, real, quickdraw, painting, and clipart domains in DomainNet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The best results are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 5-way, 1-shot Methods skt ←→ rel skt ←→ qdr skt ←→ pnt skt ←→ cli rel ←→ qdr rel ←→ pnt rel ←→ cli qdr ←→ pnt qdr ←→ cli pnt ←→ cli avg → / ← → / ← → / ← → / ← → / ← → / ← → / ← → / ← → / ← → / ← MCD 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='07/37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='74 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='74/63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='02 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='99/80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='31 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='67/61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='63 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='25 Table 3: Ablation study (%) of the modules designed in TSECS, where the FS-UDA tasks are evaluated from a do- main (sketch) to the other four domains in DomainNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Components Target Domains TSE catt CS cli rel qdr pnt ✓ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='98 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='00 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='21 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='62 ✓ 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='07 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='31 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='93 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='66 ✓ ✓ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='74 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='54 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='64 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='23 ✓ ✓ 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='25 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='15 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='31 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='34 ✓ ✓ ✓ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='45 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='00 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='25 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='51 mance becomes relatively low when the domain gap is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' For example, from quickdraw to painting, it performs the worst in all the other combinations because of larger domain gap, but our TSECS outperforms IMSE and the other com- pared methods by 8% and 12%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We found that our method has the larger performance improvement over IMSE, for these combinations containing quickdraw, which shows the efficacy of our method for large domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Also, like TSECS, IMSE+TSE performs much better than IMSE for large domain gap, which indicates the high-level seman- tic features could conduct domain adaptation better than lo- cal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In sum, these results reflect the advantages of our TSECS to deal with domain shift and task generaliza- tion in FS-UDA, no matter how large the domain gap is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Ablation study of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We conduct various ex- periments on DomainNet to evaluate the effect of our mod- ules: task-specific semantic embedding (TSE), cross-domain self-training (CS) and cross-attention in TSE (catt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The ac- curacies on the four target domains are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' As seen, our method achieve the best performance when three modules are all used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The performance of the single CS is the worst that shows that local features cannot align the distributions of the two domains, thus affecting cross- domain self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The module TSE is introduced into four combinations, all improving the performance, which validates the efficacy of our task-specific semantic features for FS-UDA again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Also, the addition of cross-attention into TSE will further improve the performance, which can help discover more semantics from previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Ablation study of different losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We conduct various experiments on DomainNet to further evaluate the effect of different losses in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Besides the classification loss (Lcls), we combine the remaining three loss terms: 1) se- mantic features alignment loss (Lsfa), 2) similarity pattern alignment loss (Lspa), and 3) class matching loss (Lclm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We evaluate 5-way 1-shot FS-UDA tasks from sketch to the other four domains, respectively, and their accuracies are re- ported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' As observed, the more the number of loss terms involved, the higher the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The combination of all the three losses is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' For the single loss, both Lsfa Table 4: Ablation study (%) of the three losses designed in TSECS, where the FS-UDA tasks are evaluated from a do- main (sketch) to the other four domains in DomainNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Components Target Domains Lsfa Lspa Lclm cli rel qdr pnt ✓ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='67 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='84 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='91 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='28 ✓ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='28 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='32 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='11 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='46 ✓ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='83 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='29 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='51 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='25 ✓ ✓ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='64 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='64 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='41 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='40 ✓ ✓ 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='04 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='98 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='13 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='39 ✓ ✓ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='61 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='47 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='07 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='14 ✓ ✓ ✓ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='45 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='00 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='25 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='51 Figure 4: Comparison of introducing our TSE module or not into two FSL methods with ADDA (Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017) com- bined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', ADDA+ProtoNet and ADDA+DN4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Lclm perform better than Lspa, and their combination is also considerably better than the other paired combinations, showing the efficacy of semantic feature domain alignment and class matching in target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Based on the above, adding Lspa further improves the performance, indicating positive effect of aligning the similarity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Evaluation on the effect of our task-specific se- mantic embedding module on two FSL methods with ADDA (Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017) combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Compared with ADDA+DN4 and ADDA+ProtoNet, we add our semantic embedding module (TSE) with the loss Lsfa into their fea- ture embedding models, and test them on 3000 new 5-way 1/5-shot FS-UDA tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' For simplification and clarification, we calculate the averaged accuracies from every domain to the other four domains and show them in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' As seen, the methods using TSE generally perform better than that without it, which validates that the semantic embedding in TSE could generate more discriminative semantic features for classification than original local features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In addition, the performances of these methods are still far from our method because using ADDA is insufficient to align the domains and could result in class mismatching, but our method can effec- tively solve it by cross-domain self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Evaluation of dataset generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We evaluate the generalization of our model trained on DomainNet to adapt to a substantially different dataset miniImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We mod- ify miniImageNet by transferring a half of real images (rel) into sketch images (skt) by MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2018) to Table 5: Evaluation (%) of dataset generalization for 5-way 1-shot FS-UDA tasks between domains real and sketch, per- forming episodic training on DomainNet and testing on ex- panded dataset miniImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Methods skt → rel rel → skt ADDA+DN4 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='87 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='90 ADDA+DeepEMD 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='82 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='77 IMSE 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='78 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='81 TSECS (ours) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='33 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='08 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='96 Figure 5: The tSNE visualization of our TSECS using cross- domain self-training or not for a 5-way 5-shot FS-UDA task from sketch to clipart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The samples with different colors be- long to different classes, and the stars in the left and right figures represent the class centroids of support set and se- lected target domain query samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' produce two domains for FS-UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We compare our method with ADDA+DN4, ADDA+DeepEMD and IMSE for 5-way 1-shot FS-UDA tasks for rel ↔ skt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' The results are shown as Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' As observed, our method outperforms other methods, specially for ske → rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' For rel → skt, our method is slightly better than IMSE, because the style of sketch im- ages in miniImageNet is relatively different from that in Do- mainNet, which could effect the learned semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Visualization of our method using cross-domain self- training or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We illustrate the tSNE results of a 5-way 5- shot FS-UDA task from sketch to clipart in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Note that the class prototypes in the left subfigure belong to the sup- port set in source domain, while those in the right subfigure are generated by ‘confidence’ samples in target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' It is obvious that two class prototypes in the left subfigure are fully overlapped so that many samples could not be correctly classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In contrast, the right subfigure has the better class prototypes, and samples from different classes are more dis- tinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' This shows the efficacy of our cross-domain self-training that finds ‘confidence’ samples to train the tar- get domain classifier and uses class matching loss Lclm to shorten the distance of samples of the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Conclusion In this paper, we propose a novel method TSECS for FS- UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' We extract high-level semantic features than local fea- tures to measure the similarity of query images in target do- main to support classes in source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Moreover, we de- sign cross-domain self-training to train a target domain clas- sifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In addition, asymmetrical KL-divergence is used to align the semantic features between domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Extensive ex- periments on DomainNet show the efficacy of our TSECS, significantly improving the performance for FS-UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ProtoNet/1-shot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='- DN4/1-shot 701 701 60 60 50 50 40 40 30 30 skt cli rel qdr pnt skt clirel qdr pnt ProtoNet/5-shot DN4/5-shot 70 70 60 60 50 50 40 40 30 30 skt clirel Ipb pnt skt clirel qdr pnt ADDA+ProtoNet ADDA+DN4 ADDA+TSE+ProtoNet ADDA+TSE+DN4 ours oursTSECS (no CS) TSECS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='0Acknowledgments Wanqi Yang and Ming Yang are supported by Na- tional Natural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 62076135, 62276138, 61876087).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Lei Wang is supported by an Australian Research Council Discovery Project (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' DP200101289) funded by the Australian Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' References Bertinetto, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Henriques, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Torr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Vedaldi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Meta-learning with differentiable closed-form solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In International Conference on Learning Represen- tations, 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Kira, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' A Closer Look at Few-shot Classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In International Conference on Learning Representa- tions, 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Finn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Abbeel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Precup, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Teh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, 1126–1135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Ganin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Ustinova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Ajakan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Germain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Larochelle, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Laviolette, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' March, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Lempitsky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Domain-Adversarial Training of Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Journal of Machine Learning Research, 17(59): 1–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Gidaris, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Bursuc, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Komodakis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Perez, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Cord, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Learning Representations by Predicting Bags of Visual Words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Confer- ence on Computer Vision and Pattern Recognition (CVPR), 6926–6936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Few-Shot Unsupervised Domain Adaptation with Image-to-Class Sparse Similarity Encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Proceedings of the 29th ACM International Conference on Multimedia, MM ’21, 677–685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' New York, NY, USA: Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ISBN 9781450386517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Belongie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Kautz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Multimodal Unsupervised Image-to-Image Translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Ferrari, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Hebert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Sminchisescu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Weiss, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', Computer Vision – ECCV 2018, 179–196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Cham: Springer International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ISBN 978-3-030-01219- 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Saito, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Oh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Plummer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Sclaroff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Saenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' CDS: Cross-Domain Self-supervised Pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In 2021 IEEE/CVF International Conference on Computer Vision (ICCV), 9103–9112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Huo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Asymmetric Distribution Measure for Few-shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Bessiere, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', Proceedings of the Twenty-Ninth Interna- tional Joint Conference on Artificial Intelligence, IJCAI-20, 2957–2963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' International Joint Conferences on Artificial In- telligence Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Main track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Huo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Revisiting Local Descriptor Based Image-To-Class Measure for Few-Shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 7260–7268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Long, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Jordan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Learn- ing Transferable Features with Deep Adaptation Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Bach, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Blei, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', Proceedings of the 32nd In- ternational Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, 97–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Lille, France: PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Luo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Guan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Ad- versarial Style Mining for One-Shot Unsupervised Domain Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Larochelle, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Ranzato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Hadsell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Balcan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', Advances in Neural Informa- tion Processing Systems, volume 33, 20612–20623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Peng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Bai, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Xia, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Saenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Moment Matching for Multi-Source Do- main Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In 2019 IEEE/CVF International Confer- ence on Computer Vision (ICCV), 1406–1415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Ravi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Larochelle, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Optimization as a Model for Few-Shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Roy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Siarohin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Sangineto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Bulo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Sebe, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Ricci, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Unsupervised Domain Adaptation Using Feature-Whitening and Consensus Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 9471–9480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Saito, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Watanabe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Ushiku, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Harada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Maximum Classifier Discrepancy for Unsupervised Domain Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 3723– 3732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Snell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Swersky, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Zemel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Prototypical Net- works for Few-shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Guyon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Luxburg, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Bengio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Fergus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Vishwanathan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Garnett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', Advances in Neural Information Process- ing Systems, volume 30, 4077–4087.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Tang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Jia, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Unsupervised Domain Adaptation via Structurally Regularized Deep Clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vi- sion and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Tseng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Cross-Domain Few-Shot Classification via Learned Feature-Wise Transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Tzeng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Hoffman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Saenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Darrell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Adversarial Discriminative Domain Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In 2017 IEEE Conference on Computer Vision and Pattern Recog- nition (CVPR), 2962–2971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Tzeng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Hoffman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Zhang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Saenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Darrell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Deep Domain Confusion: Maximizing for Domain Invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' CoRR, abs/1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='3474: 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Vinyals, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Blundell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Lillicrap, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' kavukcuoglu, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Wierstra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Matching Networks for One Shot Learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Sugiyama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Luxburg, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Guyon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Garnett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', Advances in Neural Information Process- ing Systems, volume 29, 3630–3638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Few-shot Unsupervised Domain Adaptation via Meta Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In IEEE International Conference on Multimedia and Expo (ICME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Ye, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Zhan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Sha, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Few-Shot Learning via Embedding Adaptation With Set-to-Set Func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Com- puter Vision and Pattern Recognition (CVPR), 8805–8814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Yue, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Zheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Darrell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Keutzer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Vincentelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Prototypical Cross-Domain Self-Supervised Learning for Few-Shot Unsupervised Do- main Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Confer- ence on Computer Vision and Pattern Recognition (CVPR), 13834–13844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Cai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Lin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Shen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Deep- EMD: Few-Shot Image Classification With Differentiable Earth Mover’s Distance and Structured Classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Pro- ceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 12200–12210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Zou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Yu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Kumar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Confidence Regularized Self-Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In 2019 IEEE/CVF International Conference on Computer Vision (ICCV), 5981–5990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Zou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Yu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Vijaya Kumar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Unsupervised Domain Adaptation for Semantic Segmenta- tion via Class-Balanced Self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' In Ferrari, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Hebert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Sminchisescu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' and Weiss, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=', Computer Vision – ECCV 2018, 297–313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' Cham: Springer International Pub- lishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} +page_content=' ISBN 978-3-030-01219-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQf__9t/content/2301.01956v1.pdf'} diff --git a/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf b/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9fdf32a85f0ccd24155a6b2044862f02eeaf0a04 --- /dev/null +++ b/BNFIT4oBgHgl3EQf_iwr/content/2301.11415v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f32181a5d2da7bab9db35759db218937e50e88ebfddf794e2127a8752a3a96ee +size 480537 diff --git a/BNFIT4oBgHgl3EQf_iwr/vector_store/index.faiss 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+PERCEPTUAL–NEURAL–PHYSICAL SOUND MATCHING +Han Han, Vincent Lostanlen, and Mathieu Lagrange +Nantes Universit´e, ´Ecole Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France +ABSTRACT +Sound matching algorithms seek to approximate a target waveform +by parametric audio synthesis. Deep neural networks have achieved +promising results in matching sustained harmonic tones. However, +the task is more challenging when targets are nonstationary and inhar- +monic, e.g., percussion. We attribute this problem to the inadequacy +of loss function. On one hand, mean square error in the parametric +domain, known as “P-loss”, is simple and fast but fails to accommo- +date the differing perceptual significance of each parameter. On the +other hand, mean square error in the spectrotemporal domain, known +as “spectral loss”, is perceptually motivated and serves in differen- +tiable digital signal processing (DDSP). Yet, spectral loss has more +local minima than P-loss and its gradient may be computationally +expensive; hence a slow convergence. Against this conundrum, we +present Perceptual-Neural-Physical loss (PNP). PNP is the optimal +quadratic approximation of spectral loss while being as fast as P-loss +during training. We instantiate PNP with physical modeling synthesis +as decoder and joint time–frequency scattering transform (JTFS) as +spectral representation. We demonstrate its potential on matching +synthetic drum sounds in comparison with other loss functions. +Index Terms— auditory similarity, scattering transform, deep +convolutional networks, physical modeling synthesis. +1. INTRODUCTION +Given an audio synthesizer g, the task of sound matching [1] consists +in retrieving the parameter setting θ that “matches” a target sound +x; i.e., such that a human ear judges the generated sound g(θ) to +resemble x. Sound matching has applications in automatic music +transcription, virtual reality, and audio engineering [2, 3]. Of particu- +lar interest is the case where g(θ) solves a known partial differential +equation (PDE) whose coefficients are contained in the vector θ. In +this case, θ reveals some key design choices in acoustical manufac- +turing, such as the shape and material properties of the resonator. +Over the past decade, the renewed interest for deep neural net- +works (DNN’s) in audio content analysis has led researchers to formu- +late sound matching as a supervised learning problem [4]. Intuitively, +the goal is to optimize the synaptic weights W of a DNN f W so +that f W(xn) = ˜θn approximates θn over a training set of pairs +(xn, θn). Because g automates the mapping from parameter θn to +sound xn, this training procedure incurs no real-world audio acquisi- +tion nor human annotation. However, prior publications have pointed +out that the approximation formula ˜θn ≈ θn lacks a perceptual mean- +ing: depending on the choice of target xn, some deviations (˜θn−θn) +may be judged to have a greater effect than others [5, 6, 7]. +The paradigm of differentiable digital signal processing (DDSP) +has brought a principled methodology to address this issue [8]. The +key idea behind DDSP is to chain the learnable encoder f W with +the known decoder g and a non-learnable but differentiable feature +map Φ. In DDSP, f W is trained to minimize the perceptual distance +θ +Parametric +domain +x +original +Audio +domain +S +Perceptual +domain +˜θ +˜x +reconstruction +˜S +DDSP +spectral ≈ +loss +PNP +quadratic +form +θ +x +˜θ +M(θ) +Riemannian +metric +g +Φ +f W +g +Φ +g +f W +∇(Φ◦g) +Fig. 1. Graphical outline of the proposed method. Given a known +synthesizer g and feature map Φ, we train a neural network f W to +minimize the “perceptual–neural–physical” (PNP) quadratic form +⟨˜θ − θ +��M(θ)|˜θ − θ⟩ where M is the Riemannian metric associated +to (Φ◦g). Hence, PNP approximates DDSP spectral loss yet does not +need to backpropagate ∇(Φ◦g)(˜θ) at each epoch. Transformations in +solid (resp. dashed) lines can (resp. cannot) be cached during training. +between vectors Φ(˜xn) = (Φ◦g◦f W)(xn) and Φ(xn) on average +over samples xn. Yet, a practical shortcoming of DDSP is that it +requires to evaluate the Jacobian ∇(Φ◦g) over each DNN prediction +˜θn; and so at every training step, as W is updated by stochastic +gradient descent (SGD). +In this article, we propose a new learning objective for sound +matching, named perceptual–neural–physical (PNP) autoencoding. +The main contribution of PNP is to compute the Riemannian met- +ric M associated to the Jacobian ∇(Φ◦g) over each sample θn (see +Section 2.1). The PNP encoder is penalized in terms of a first-order +Taylor expansion of the spectral loss ∥Φ(˜x) − Φ(x)∥2, making +it comparable to DDSP. Yet, unlike in DDSP, the computation of +∇(Φ◦g) is independent from the encoder f W: thus, it may be par- +allelized and cached during DNN training. A second novelty of +our paper resides in its choice of application: namely, differentiable +sound matching for percussion instruments. This requires not only +a fine characterization of the spectral envelope, as in the DDSP of +sustained tones; but also of attack and release transients. For this +purpose, we need g and Φ to accommodate sharp spectrotemporal +modulations. Specifically, we rely on a differentiable implementation +of the functional transformation method (FTM) for g and the joint +time–frequency scattering transform (JTFS) for Φ. +2. METHODS +2.1. Approximating spectral loss with Riemannian geometry +We assume the synthesizer g and the feature map Φ to be contin- +uously differentiable. Let us denote by LDDSP the “spectral loss” +arXiv:2301.02886v1 [cs.SD] 7 Jan 2023 + +associated to the triplet (Φ, f W, g). Its value at a parameter set θ is: +LDDSP +θ +(W) = 1 +2∥Φ(˜x) − Φ(x)∥2 +2 += 1 +2 +��(Φ ◦ g ◦ f W ◦ g)(θ) − (Φ ◦ g)(θ) +��2 +2 +(1) +by definition of ˜x and x. Using ˜θ as shorthand for (f W ◦ g)(θ), we +conduct a first-order Taylor expansion of (Φ ◦ g) near θ. We obtain: +Φ(˜x) = Φ(x) + ∇(Φ◦g)(θ) · (˜θ − θ) + O(∥˜θ − θ∥2 +2), +(2) +where the Jacobian matrix ∇(Φ◦g)(θ) contains P = dim Φ(x) rows +and J = dim θ columns. The differentiable map (Φ ◦ g) induces a +weak Riemannian metric M onto the open set U ⊂ RJ of parameters +θ, whose matrix values derive from ∇(Φ◦g)(θ): +M(θ)j,j′ = +P +� +p=1 +� +∇(Φ◦g)(θ)p,j +� � +∇(Φ◦g)(θ)p,j′� +. +(3) +The square matrix M(θ) ∈ GLJ(R) defines a positive semidefinite +kernel which, once plugged into Equation 2, serves to approximate +LDDSP +θ +(W) in terms of a quadratic form over (˜θ − θ): +∥Φ(˜x) − Φ(x)∥2 +2 = +�˜θ − θ +��M(θ) +��˜θ − θ +� ++ O +� +∥˜θ − θ∥3 +2 +� +. (4) +The advantage of the approximation above is that the metric +M may be computed over the training set once and for all. This is +because Equation 3 is independent of the encoder f W. Furthermore, +since θ is low-dimensional, we may store M(θ) on RAM. From +this perspective, we define the perceptual–neural–physical loss (PNP) +associated to (Φ, f W, g) as the linearization of spectral loss at θ: +LPNP +θ +(W) = 1 +2 +� +(f W ◦ g)(θ) − θ +��M(θ) +��(f W ◦ g)(θ) − θ +� += LDDSP +θ +(W) + O +� +∥(f W ◦ g)(θ) − θ∥3 +2 +� +. +(5) +According to the chain rule, the gradient of PNP loss at a given +training pair (xn, θn) with respect to some scalar weight Wi is: +∂LPNP +θ +∂Wi (θn) = +� +f W(xn) − θn +���M(θn) +���∂f W +∂Wi (xn) +� +. +(6) +Observe that replacing M(θn) by the identity matrix in the equation +above would give the gradient of parameter loss (P-loss); that is, +the mean squared error between the predicted parameter ˜θ and the +true parameter θ. Hence, we may regard PNP as a perceptually +motivated extension of P-loss, in which parameter deviations are +locally recombined and rescaled so as to simulate a DDSP objective. +The matrix M(θ) is constant in W. Hence, its value may be +cached across training epochs, and even across hyperparameter set- +tings of the encoder. In comparison with P-loss, the only computa- +tional overhead of PNP is the bilinear form in Equation 6. However, +this computation is performed in the parametric domain, i.e., in low +dimension (J = dim θ). Hence, its cost is negligible in front of the +forward (f W) and backward pass (∂f W/∂Wi) of DNN training. +2.2. Damped least squares +The principal components of the Jacobian ∇(Φ◦g)(θ) are the eigen- +vectors of M(θ). We denote them by vj and the corresponding +eigenvalues by σ2 +j : for each of them, we have M(θ)vj = σ2 +j vj. The +vj’s form an orthonormal basis of RJ, in which we can decompose +the parameter deviation (˜θ − θ). Recalling Equation 5, we obtain an +alternative formula for PNP loss: +LPNP +θ +(W) = 1 +2 +J +� +j=1 +σ2 +j +��⟨(f W ◦ g)(θ) − θ +��vj⟩ +��2 v2 +j +(7) +The eigenvalues σ2 +j stretch and compress the error vector along their +associated direction vj, analogous to the magnification and suppres- +sion of perceptually relevant and irrelevant parameter deviations. In +practice however, when σ2 +j cover drastic ranges or contain zeros, as +presented below in Section 4.3, the error vector is subject to extreme +distortion and potential instability due to numerical precision errors. +These scenarios, commonly referred to as M being ill-conditioned, +can lead to intractable learning objective LPNP +θ +. +Reminiscent of the damping mechanism introduced in Levenberg- +Marquardt algorithm when solving nonlinear optimization problems, +we update Equation 5 as +LPNP +θ +(W) = 1 +2 +�˜θ − θ +��M(θ) + λI +��˜θ − θ +� +(8) +The damping term λI up-shifts all eigenvalues of M by a constant +positive amount λ, thereby changing its condition number. When +λ is huge, M(θ) + λI is close to an identity matrix with uniform +eigenvalues, LPNP +θ +is optimizing in parameter loss regime. On the +other hand when λ is small, small correctional effects keeps LPNP +θ +in the spectral loss regime. Alternatively, Equation 8 may also be +viewed as a L2 regularization with coefficient λ, which allows smooth +transition between spectral and parameter loss regimes. +To further address potential convergence issues, λ may be sched- +uled or adaptively changed according to epoch validation loss. We +adopt delayed gratification mechanism to decrease λ by a factor of 5 +when loss is going down, and fix λ otherwise. +3. APPLICATION TO DRUM SOUND MATCHING +3.1. Perceptual: Joint time–frequency scattering (JTFS) +The joint time–frequency scattering transform (JTFS) is a nonlinear +convolutional operator which extracts spectrotemporal modulations +in the constant-Q scalogram [9]. Its kernels proceed from a separable +product between two complex-valued wavelet filterbanks, defined +over the time axis and over the log-frequency axis respectively. After +convolution, we apply pointwise complex modulus and temporal +averaging to each JTFS coefficient. These coefficients are known as +scattering “paths” p. We apply a logarithmic transformation to the +feature vector JTFS(xn) corresponding to each sound xn, yielding +Sn,p = (Φ ◦ g)(θn)p = log +� +1 + JTFS(xn)p +ε +� +, +(9) +where we have set the hyperparameter ε = 10−3 of the order of the +median value of JTFS across all examples xn and paths p. +The multiresolution structure of JTFS is reminiscent of spec- +trotemporal receptive fields (STRF), and thus may serve as a bio- +logically plausible predictor of neurophysiological responses in the +primary auditory cortex [10]. At a higher level of music cognition, a +recent study has shown that Euclidean distances in Φ space predict +auditory judgments of timbre similarity within a large vocabulary +of instrumental playing techniques, as collected from a group of +professional composers and non-expert music listeners [11]. +We compute JTFS with same parameters as [11]: Q1 = 12, +Q2 = 1, and Qfr = 1 filters per octave respectively. We set the + +temporal averaging to T = 3 seconds and the frequential averaging +to F = 2 octaves; hence a total of P = 20762 paths. We run Φ and +∇(Φ◦g) in PyTorch on GPU via the implementation of [12, 13]. +3.2. Neural: Deep convolutional network (convnet) +EfficientNet is a convolutional neural network architecture that bal- +ances the scaling of the depth, width and input resolution of con- +secutive convolutional blocks [14]. Achieving state-of-the-art per- +formance on image classification with significantly less trainable +parameters, its most light-weight version EfficientNet-B0 also suc- +ceeded in benchmarking audio classification tasks [15]. We adopt +EfficientNet-B0 as our encoder f W, resulting in 4M learnable pa- +rameters. We append a linear dense layer of J = dim θ neurons +and a 1D batch normalization before tanh activation. The goal of +batch normalization is to gaussianize the input, such that the activated +output is capable of uniformly cover the normalized prediction range. +The input to f W is the log-scaled CQT coefficients of each example, +computed with a filterbank spanning 10 octaves with 12 filters per +octave. +3.3. Physical: Functional transformation method (FTM) +We are interested in the perpendicular displacement X(t, u) on a +rectangular drum face, which can be solved from the following partial +differential equation defined in the Cartesian coordinate system u = +(u1, u2). +�∂2X +∂t2 (t, u) − c2∇2X(t, u) +� ++ S4� +∇4X(t, u) +� ++ ∂ +∂t +� +d1X(t, u) + d3∇2X(t, u) +� += Y(t, u) +(10) +In addition to the standard traveling wave equation in the first above +parenthesis, the fourth-order spatial and first-order time derivatives +incorporate damping factors induced by stiffness, internal friction +in the drum material and air friction in the external environment, +rendering the solution a closer simulation to reality. Specifically, +α, S, c, d1, d3 designate respectively the side length ratio, stiffness, +traveling wave speed, frequency-independent damping and frequency- +dependent damping of the drum. Even though real world drums +are mostly circular, a rectangular drum model is equally capable of +eliciting representative percussive sounds in real world scenarios. The +circular drum model simply requires a conversion of Equation 10 +into the Polar coordinate system. We bound the four sides of this l +by lα rectangular drum at zero at all time. Moreover, we assume its +excitation function to be separable and localized in space and time +Y(t, u) = yu(u)δ(t). +We implement generator g as a PDE solver to this high-order +damped wave equation, namely the functional transformation method +(FTM) [16, 17]. FTM solves the PDE by transforming the equation +into its Laplace and functional space domain, where an algebraic +solution can be obtained. It then finds the time-space domain solution +via inverse functional transforms, expressed in an infinite modal +summation form +x(t) = X(t, u) = +� +m∈N2 +Km(u, t) exp(σmt) sin(ωmt) +(11) +The coefficients Km(u, t), σm, ωm are derived from the original +PDE parameters in the following ways. +ω2 +m = (S4 − d2 +3 +4 )Γ2 +m1,m2 + (c2 + d1d3 +2 +)Γm1,m2 − d2 +1 +4 +(12) +Fig. 2. Distributions of the sorted eigenvalues of M(θn). For the +sake of comparison between PNP and P-loss, the dashed line indicates +the eigenvalues of the identity matrix (see Equation 6). +σm = d3 +2 Γm1,m2 − d1 +2 +(13) +Km(u, t) = ym +u δ(t) sin(πm1u1 +l +) sin +�πm2u2 +lα +� +(14) +where Γm1,m2 = π2m2 +1/l2 + π2m2 +2/(lα)2, and ym +u is the mth coef- +ficient associated to the eigenfunction sin(πmu/l) that decomposes +yu(u). +Without losing connections to the acoustical manufacturing of +the drum yet better relating g’s input with perceptual dimensions, +we reparametrize the PDE parameters {S, c, d1, d3, α} into θ = +{log ω1, τ1, log p, log D, α}, detailed in Section 3.4 of [18]. We pre- +scribe sonically-plausible ranges for each parameter in θ, normalize +them between −1 and 1, uniformly sample in the hyper-dimensional +cube, and obtain a dataset of 100k percussive sounds sampled at +22050 HZ. The train/test/validation split is 8 : 1 : 1. +In particular, fundamental frequency ω1, duration τ1 falls into +ranges [40, 1000] Hz and [0.4, 3] seconds respectively. Inhomoge- +neous damping rate p, frequential dispersion D and aspect ratio α +ranges are [10−5, 0.2], [10−5, 0.3], and [10−5, 1]. +4. RESULTS +4.1. Baselines +We train fW with 3 different losses - multi-scale spectral loss [19], +parameter loss, and PNP loss. We use a batch size of 64 samples +for spectral loss, and 256 samples for parameter and PNP loss. The +training proceeds for 70 epochs, where around 20% of the training +set is seen at each epoch. We use Adam optimizer with learning rate +10−3. Table 1 reports the training time per epoch on a single Tesla +V100 16GB GPU. +4.2. Evaluation with JTFS-based spectral loss +We propose to use the L2 norm of JTFS coefficients error averaged +over test set for evaluation. As a point of reference, we also include +the average multi-scale spectral error, implemented as in Section +4.1. One of the key distinctions between Euclidean JTFS distance +and multi-scale spectral error is the former’s inclusion of spectro- +temporal modulations information. Meanwhile unlike mean squared +parameter error, both metrics reflect the perceptual closeness instead +of parametric retrieval accuracy for each proposed model. +4.3. Discussion +Despite being the optimal quadratic approximation of spectral loss, +it is nontrivial to apply the bear PNP loss form as Equation 5 in + +OCD +15 +10 +5 +0 +5 +log1o(on,j)Pitch +JTFS distance +(avg. on test set) +MSS +(avg. on test set) +Training time +per epoch +P-loss +Known +22.23 ± 2.17 +0.31 ± 0.013 +49 minutes +DDSP with MSS loss +Known +31.86 ± 0.332 +0.335 ± 0.005 +54 minutes +PNP with JTFS loss +Known +23.58 ± 0.877 +0.335 ± 0.005 +49 minutes +DDSP with JTFS loss +— +— +— +est., > 1 day +P-loss +Unknown +61.91 ± 6.26 +1.02 ± 0.094 +53 minutes +DDSP with MSS loss +Unknown +138.95 ± 37.12 +1.59 ± 0.307 +59 minutes +PNP with JTFS loss +Unknown +61.21 ± 1.207 +0.97 ± 0.019 +49 minutes +Table 1. Report of average JTFS distance and MSS metrics evaluated on test set. Six models are trained with two modalities: 1. the inclusion +of pitch retrieval i.e.regressing θ = {τ, log p, log D, α} vs. θ = {log ω1, τ, log p, log D, α}, and 2. the choice of loss function: P-loss, MSS +loss, or PNP loss with adaptive damping mechanism. The best performing models with known and unknown pitch are P-loss and PNP loss +respectively. Training with MSS loss is more time consuming than training with P-loss or PNP loss. Training with differentiable JTFS loss is +unrealistic in the interest of time. +experimental settings. On one hand, Φ◦g potentially has undesirable +property that exposes the Riemannian metric calculations to numeri- +cal precision errors. On the other hand, extreme deformation of the +optimization landscape may lead to the same numerical instability +facing stochastic gradient descent with spectral loss. We report on a +few remedies that helped stabilize learning with PNP loss, and offer +insights on future directions to take. +First and foremost, our preliminary experiments show that train- +ing PNP loss without damping λ = 0 subjects to serious convergence +issues. Recalling Section 2.2, indeed our empirical Ms suffer from +high condition numbers. Fig. 2 shows the sorted eigenvalue distribu- +tion of all Ms in test set, where Ms are rank-2,3 or 4 matrices with +eigenvalues ranging from 0 to 1020. This could be an implication +that entries of θ contain implicit linear dependencies in generator g, +or that local variations of certain θ fail to linearize differences in the +output of g or Φ ◦ g. As an example, the aspect ratio α influences the +modal frequencies and decay rates via [18, Equations 12–13], where +in fact its variant 1/α + 1/α2 could be a better choice of variable +that linearizes g. +To address Ms’ ill conditions we attempted at numerous damping +mechanisms to update λ, namely constant λ, scheduled λ decay, and +adaptive λ decay. The intuition is to have LPNP +θ +start in the parameter +loss regime and move towards the spectral loss regime while training. +The best performing model is achieved with adaptive λ decay, as +described in Section 2.2. We initialize λ to be 1020 to match the +largest empirical σ2 +j , which later gets adaptively decayed to 3 × 1014 +in 20 epochs, breaking records 7 times. This indicates that f W is +able to learn with damped PNP loss, under the condition that λ being +large enough to simulate parameter loss regime and compensate for +deficiency in M. +The diagonal elements of M(θ) can be regarded as both the ap- +plied weights’ magnitudes and proxies for θ’s perceptual significance. +To gain further insights on how each model regresses different pa- +rameters, we visualize in Fig.3 pairs of (|˜θ − θ|2 +j, M(θ)j,j). Three +trends can be observed: First, τ and ω are regressed with the best +accuracy across all learning objectives. Second, spectral loss particu- +larly struggles in pitch ω and inharmonicity p retrievals. Third, we +may interpret x-axis as describing from left to right samples with +increasing perceptual significance. We observe that in Fig.3(b), PNP +loss is able to suppress more errors in samples with high M(θ)j,j +than parameter loss, by a nonnegligible margin. +We believe that more of PNP loss’ mathematical potential can +be exploited in the future, notably its ability to interpolate between +various loss regimes and its use in hybrid optimization schemes. To +start with, we plan to resort to a simpler differentiable synthesizer g +Fig. 3. X-axis: weight assigned by PNP to one of the physical +parameters in θn. Y-axis: log squared estimation error for that same +parameter. α is omitted due to its poor retrieval results from all +models. +that guarantees a well-conditioned Riemannian metric M(θ). More- +over, we plan to explore other damping schemes and optimizers. The +current update mechanism, originated from the Leverberg-Marquardt +Algorithm, aims to improve the conditioning of a matrix inversion +problem in the Gauss-Newton algorithm. However when used jointly +with stochastic gradient descent, each λ update may change the opti- +mization landscape drastically. The resulting optimization behavior +is thus not fully understood. We consider interfacing nonlinear least +squares solver with SGD and forming a hybrid learning scheme in +future work. +5. CONCLUSION +In this article we have presented Perceptual-Neural-Physical (PNP) +autoencoding, a bilinear form learning objective for sound matching +task. In our application, PNP optimizes the retrieval of physical +parameters from sounds in a perceptually-motivated metric space, +enabled by differentiable implementations of domain knowledge in +physical modeling and computational proxy of neurophysiological +construct of human auditory system. +We demonstrated PNP’s mathematical proximity to spectral loss +and its generalizability to parameter loss. Using this formulation, +we motivated and established one way of enabling smooth transition +between optimizing in parameter and spectral loss regimes. We +have presented damping mechanisms to facilitate its learning under +ill-conditioned empirical settings and discussed its mathematical +potential. + +w - w]2 vs. M[0,0] + - T|2 vs. M[1, 1] +100 +10~2 +10-3 +10~5 +Ploss +10~6 +10-8 +Spec +109 +PNP +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +100000 +200000 +300000 +400000 +(a) +le10 +(b) +Ip -p]2 vs. M[2,2] +ID - D2 vs. M[3,3] +101 +100 +10~3 +10~2 +10~6, +10-5 +10-9 +10-8 +0 +50000 +100000 +150000 +200000 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +(c) +(d) +le106. REFERENCES +[1] Andrew Horner, “Wavetable matching synthesis of dynamic +instruments with genetic algorithms,” Journal of the Audio +Engineering Society, vol. 43, no. 11, pp. 916–931, 1995. +[2] Jordie Shier, Kirk McNally, George Tzanetakis, and Ky Grace +Brooks, +“Manifold learning methods for visualization and +browsing of drum machine samples,” Journal of the Audio +Engineering Society, vol. 69, no. 1/2, pp. 40–53, 2021. +[3] Philippe Esling, Naotake Masuda, Adrien Bardet, Romeo De- +spres, Axel Chemla, et al., “Universal audio synthesizer control +with normalizing flows,” in Proceedings of the International +Conference on Digital Audio Effects (DAFX), 2019. +[4] Leonardo Gabrielli, Stefano Tomassetti, Carlo Zinato, and +Francesco Piazza, +“End-to-end learning for physics-based +acoustic modeling,” IEEE Transactions on Emerging Topics in +Computational Intelligence, vol. 2, no. 2, pp. 160–170, 2018. +[5] Naotake Masuda and Daisuke Saito, “Synthesizer sound match- +ing with differentiable DSP.,” in Proceedings of the Interna- +tional Society on Music Information Retrieval (ISMIR) Confer- +ence, 2021, pp. 428–434. +[6] Martin Roth and Matthew Yee-king, “A comparison of para- +metric optimization techniques for musical instrument tone +matching,” Journal of the Audio Engineering Society, May +2011. +[7] Matthew Yee-King, Leon Fedden, and Mark d’Inverno, “Au- +tomatic programming of vst sound synthesizers using deep +networks and other techniques,” IEEE Transactions on Emerg- +ing Topics in Computational Intelligence, vol. 2, pp. 150–159, +04 2018. +[8] Jesse Engel, Lamtharn (Hanoi) Hantrakul, Chenjie Gu, and +Adam Roberts, “DDSP: Differentiable Digital Signal Process- +ing,” in Proceedings of the International Conference on Learn- +ing Representations (ICLR), 2020. +[9] Joakim And´en, Vincent Lostanlen, and St´ephane Mallat, “Joint +time–frequency scattering,” IEEE Transactions on Signal Pro- +cessing, vol. 67, no. 14, pp. 3704–3718, 2019. +[10] Taishih Chi, Powen Ru, and Shihab A Shamma, “Multiresolu- +tion spectrotemporal analysis of complex sounds,” The Journal +of the Acoustical Society of America, vol. 118, no. 2, pp. 887– +906, 2005. +[11] Vincent Lostanlen, Christian El-Hajj, Mathias Rossignol, +Gr´egoire Lafay, Joakim And´en, and Mathieu Lagrange, “Time– +frequency scattering accurately models auditory similarities +between instrumental playing techniques,” EURASIP Journal +on Audio, Speech, and Music Processing, vol. 2021, no. 1, pp. +1–21, 2021. +[12] Mathieu Andreux, Tom´as Angles, Georgios Exarchakis, +Roberto Leonarduzzi, Gaspar Rochette, Louis Thiry, John +Zarka, St´ephane Mallat, Joakim And´en, Eugene Belilovsky, +Joan Bruna, Vincent Lostanlen, Muawiz Chaudhary, Matthew J. +Hirn, Edouard Oyallon, Sixin Zhang, Carmine Cella, and +Michael Eickenberg, +“Kymatio: Scattering transforms in +Python.,” +Journal of Machine Learning Research, vol. 21, +no. 60, pp. 1–6, 2020. +[13] John Muradeli, Cyrus Vahidi, Changhong Wang, Han Han, Vin- +cent Lostanlen, Mathieu Lagrange, and George Fazekas, “Dif- +ferentiable time-frequency scattering in kymatio,” in Proceed- +ings of the International Conference on Digital Audio Effects +(DAFX), 2022. +[14] Mingxing Tan and Quoc Le, “EfficientNet: Rethinking model +scaling for convolutional neural networks,” in Proceedings +of the International conference on Machine Learning (ICML). +PMLR, 2019, pp. 6105–6114. +[15] Neil Zeghidour, Olivier Teboul, F´elix de Chaumont Quitry, and +Marco Tagliasacchi, “Leaf: A learnable frontend for audio +classification,” ICLR, 2021. +[16] L. Trautmann and Rudolf Rabenstein, Digital Sound Synthesis +by Physical Modeling Using the Functional Transformation +Method, 01 2003. +[17] Maximilian Sch¨afer, Manuel Werner, and Rudolf Rabenstein, +“Physical modeling in sound synthesis: Vibrating plates,” 05 +2019. +[18] Han Han and Vincent Lostanlen, “wav2shape: Hearing the +Shape of a Drum Machine,” in Proceedings of Forum Acusticum, +2020, pp. 647–654. +[19] Christian J. Steinmetz and Joshua D. Reiss, “auraloss: Audio +focused loss functions in PyTorch,” in Digital Music Research +Network One-day Workshop (DMRN+15), 2020. + diff --git a/DdE1T4oBgHgl3EQfEAP6/content/tmp_files/load_file.txt b/DdE1T4oBgHgl3EQfEAP6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1e378120bbee3cd40c2cc61e2efbbb2d9f6573e --- /dev/null +++ b/DdE1T4oBgHgl3EQfEAP6/content/tmp_files/load_file.txt @@ -0,0 +1,313 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf,len=312 +page_content='PERCEPTUAL–NEURAL–PHYSICAL SOUND MATCHING Han Han, Vincent Lostanlen, and Mathieu Lagrange Nantes Universit´e, ´Ecole Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France ABSTRACT Sound matching algorithms seek to approximate a target waveform by parametric audio synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Deep neural networks have achieved promising results in matching sustained harmonic tones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' However, the task is more challenging when targets are nonstationary and inhar- monic, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=', percussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We attribute this problem to the inadequacy of loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' On one hand, mean square error in the parametric domain, known as “P-loss”, is simple and fast but fails to accommo- date the differing perceptual significance of each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' On the other hand, mean square error in the spectrotemporal domain, known as “spectral loss”, is perceptually motivated and serves in differen- tiable digital signal processing (DDSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Yet, spectral loss has more local minima than P-loss and its gradient may be computationally expensive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' hence a slow convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Against this conundrum, we present Perceptual-Neural-Physical loss (PNP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' PNP is the optimal quadratic approximation of spectral loss while being as fast as P-loss during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We instantiate PNP with physical modeling synthesis as decoder and joint time–frequency scattering transform (JTFS) as spectral representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We demonstrate its potential on matching synthetic drum sounds in comparison with other loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Index Terms— auditory similarity, scattering transform, deep convolutional networks, physical modeling synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' INTRODUCTION Given an audio synthesizer g, the task of sound matching [1] consists in retrieving the parameter setting θ that “matches” a target sound x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=', such that a human ear judges the generated sound g(θ) to resemble x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Sound matching has applications in automatic music transcription, virtual reality, and audio engineering [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Of particu- lar interest is the case where g(θ) solves a known partial differential equation (PDE) whose coefficients are contained in the vector θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' In this case, θ reveals some key design choices in acoustical manufac- turing, such as the shape and material properties of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Over the past decade, the renewed interest for deep neural net- works (DNN’s) in audio content analysis has led researchers to formu- late sound matching as a supervised learning problem [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Intuitively, the goal is to optimize the synaptic weights W of a DNN f W so that f W(xn) = ˜θn approximates θn over a training set of pairs (xn, θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Because g automates the mapping from parameter θn to sound xn, this training procedure incurs no real-world audio acquisi- tion nor human annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' However, prior publications have pointed out that the approximation formula ˜θn ≈ θn lacks a perceptual mean- ing: depending on the choice of target xn, some deviations (˜θn−θn) may be judged to have a greater effect than others [5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The paradigm of differentiable digital signal processing (DDSP) has brought a principled methodology to address this issue [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The key idea behind DDSP is to chain the learnable encoder f W with the known decoder g and a non-learnable but differentiable feature map Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' In DDSP, f W is trained to minimize the perceptual distance θ Parametric domain x original Audio domain S Perceptual domain ˜θ ˜x reconstruction ˜S DDSP spectral ≈ loss PNP quadratic form θ x ˜θ M(θ) Riemannian metric g Φ f W g Φ g f W ∇(Φ◦g) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Graphical outline of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Given a known synthesizer g and feature map Φ, we train a neural network f W to minimize the “perceptual–neural–physical” (PNP) quadratic form ⟨˜θ − θ ��M(θ)|˜θ − θ⟩ where M is the Riemannian metric associated to (Φ◦g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Hence, PNP approximates DDSP spectral loss yet does not need to backpropagate ∇(Φ◦g)(˜θ) at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Transformations in solid (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' dashed) lines can (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' cannot) be cached during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' between vectors Φ(˜xn) = (Φ◦g◦f W)(xn) and Φ(xn) on average over samples xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Yet, a practical shortcoming of DDSP is that it requires to evaluate the Jacobian ∇(Φ◦g) over each DNN prediction ˜θn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' and so at every training step, as W is updated by stochastic gradient descent (SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' In this article, we propose a new learning objective for sound matching, named perceptual–neural–physical (PNP) autoencoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The main contribution of PNP is to compute the Riemannian met- ric M associated to the Jacobian ∇(Φ◦g) over each sample θn (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The PNP encoder is penalized in terms of a first-order Taylor expansion of the spectral loss ∥Φ(˜x) − Φ(x)∥2, making it comparable to DDSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Yet, unlike in DDSP, the computation of ∇(Φ◦g) is independent from the encoder f W: thus, it may be par- allelized and cached during DNN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' A second novelty of our paper resides in its choice of application: namely, differentiable sound matching for percussion instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' This requires not only a fine characterization of the spectral envelope, as in the DDSP of sustained tones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' but also of attack and release transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' For this purpose, we need g and Φ to accommodate sharp spectrotemporal modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Specifically, we rely on a differentiable implementation of the functional transformation method (FTM) for g and the joint time–frequency scattering transform (JTFS) for Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Approximating spectral loss with Riemannian geometry We assume the synthesizer g and the feature map Φ to be contin- uously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Let us denote by LDDSP the “spectral loss” arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='02886v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='SD] 7 Jan 2023 associated to the triplet (Φ, f W, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Its value at a parameter set θ is: LDDSP θ (W) = 1 2∥Φ(˜x) − Φ(x)∥2 2 = 1 2 ��(Φ ◦ g ◦ f W ◦ g)(θ) − (Φ ◦ g)(θ) ��2 2 (1) by definition of ˜x and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Using ˜θ as shorthand for (f W ◦ g)(θ), we conduct a first-order Taylor expansion of (Φ ◦ g) near θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We obtain: Φ(˜x) = Φ(x) + ∇(Φ◦g)(θ) · (˜θ − θ) + O(∥˜θ − θ∥2 2), (2) where the Jacobian matrix ∇(Φ◦g)(θ) contains P = dim Φ(x) rows and J = dim θ columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The differentiable map (Φ ◦ g) induces a weak Riemannian metric M onto the open set U ⊂ RJ of parameters θ, whose matrix values derive from ∇(Φ◦g)(θ): M(θ)j,j′ = P � p=1 � ∇(Φ◦g)(θ)p,j � � ∇(Φ◦g)(θ)p,j′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' (3) The square matrix M(θ) ∈ GLJ(R) defines a positive semidefinite kernel which, once plugged into Equation 2, serves to approximate LDDSP θ (W) in terms of a quadratic form over (˜θ − θ): ∥Φ(˜x) − Φ(x)∥2 2 = �˜θ − θ ��M(θ) ��˜θ − θ � + O � ∥˜θ − θ∥3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' (4) The advantage of the approximation above is that the metric M may be computed over the training set once and for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' This is because Equation 3 is independent of the encoder f W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Furthermore, since θ is low-dimensional, we may store M(θ) on RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' From this perspective, we define the perceptual–neural–physical loss (PNP) associated to (Φ, f W, g) as the linearization of spectral loss at θ: LPNP θ (W) = 1 2 � (f W ◦ g)(θ) − θ ��M(θ) ��(f W ◦ g)(θ) − θ � = LDDSP θ (W) + O � ∥(f W ◦ g)(θ) − θ∥3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' (5) According to the chain rule, the gradient of PNP loss at a given training pair (xn, θn) with respect to some scalar weight Wi is: ∂LPNP θ ∂Wi (θn) = � f W(xn) − θn ���M(θn) ���∂f W ∂Wi (xn) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' (6) Observe that replacing M(θn) by the identity matrix in the equation above would give the gradient of parameter loss (P-loss);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' that is, the mean squared error between the predicted parameter ˜θ and the true parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Hence, we may regard PNP as a perceptually motivated extension of P-loss, in which parameter deviations are locally recombined and rescaled so as to simulate a DDSP objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The matrix M(θ) is constant in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Hence, its value may be cached across training epochs, and even across hyperparameter set- tings of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' In comparison with P-loss, the only computa- tional overhead of PNP is the bilinear form in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' However, this computation is performed in the parametric domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=', in low dimension (J = dim θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Hence, its cost is negligible in front of the forward (f W) and backward pass (∂f W/∂Wi) of DNN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Damped least squares The principal components of the Jacobian ∇(Φ◦g)(θ) are the eigen- vectors of M(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We denote them by vj and the corresponding eigenvalues by σ2 j : for each of them, we have M(θ)vj = σ2 j vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The vj’s form an orthonormal basis of RJ, in which we can decompose the parameter deviation (˜θ − θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Recalling Equation 5, we obtain an alternative formula for PNP loss: LPNP θ (W) = 1 2 J � j=1 σ2 j ��⟨(f W ◦ g)(θ) − θ ��vj⟩ ��2 v2 j (7) The eigenvalues σ2 j stretch and compress the error vector along their associated direction vj, analogous to the magnification and suppres- sion of perceptually relevant and irrelevant parameter deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' In practice however, when σ2 j cover drastic ranges or contain zeros, as presented below in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='3, the error vector is subject to extreme distortion and potential instability due to numerical precision errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' These scenarios, commonly referred to as M being ill-conditioned, can lead to intractable learning objective LPNP θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Reminiscent of the damping mechanism introduced in Levenberg- Marquardt algorithm when solving nonlinear optimization problems, we update Equation 5 as LPNP θ (W) = 1 2 �˜θ − θ ��M(θ) + λI ��˜θ − θ � (8) The damping term λI up-shifts all eigenvalues of M by a constant positive amount λ, thereby changing its condition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' When λ is huge, M(θ) + λI is close to an identity matrix with uniform eigenvalues, LPNP θ is optimizing in parameter loss regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' On the other hand when λ is small, small correctional effects keeps LPNP θ in the spectral loss regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Alternatively, Equation 8 may also be viewed as a L2 regularization with coefficient λ, which allows smooth transition between spectral and parameter loss regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' To further address potential convergence issues, λ may be sched- uled or adaptively changed according to epoch validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We adopt delayed gratification mechanism to decrease λ by a factor of 5 when loss is going down, and fix λ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' APPLICATION TO DRUM SOUND MATCHING 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Perceptual: Joint time–frequency scattering (JTFS) The joint time–frequency scattering transform (JTFS) is a nonlinear convolutional operator which extracts spectrotemporal modulations in the constant-Q scalogram [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Its kernels proceed from a separable product between two complex-valued wavelet filterbanks, defined over the time axis and over the log-frequency axis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' After convolution, we apply pointwise complex modulus and temporal averaging to each JTFS coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' These coefficients are known as scattering “paths” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We apply a logarithmic transformation to the feature vector JTFS(xn) corresponding to each sound xn, yielding Sn,p = (Φ ◦ g)(θn)p = log � 1 + JTFS(xn)p ε � , (9) where we have set the hyperparameter ε = 10−3 of the order of the median value of JTFS across all examples xn and paths p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The multiresolution structure of JTFS is reminiscent of spec- trotemporal receptive fields (STRF), and thus may serve as a bio- logically plausible predictor of neurophysiological responses in the primary auditory cortex [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' At a higher level of music cognition, a recent study has shown that Euclidean distances in Φ space predict auditory judgments of timbre similarity within a large vocabulary of instrumental playing techniques, as collected from a group of professional composers and non-expert music listeners [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We compute JTFS with same parameters as [11]: Q1 = 12, Q2 = 1, and Qfr = 1 filters per octave respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We set the temporal averaging to T = 3 seconds and the frequential averaging to F = 2 octaves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' hence a total of P = 20762 paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We run Φ and ∇(Φ◦g) in PyTorch on GPU via the implementation of [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Neural: Deep convolutional network (convnet) EfficientNet is a convolutional neural network architecture that bal- ances the scaling of the depth, width and input resolution of con- secutive convolutional blocks [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Achieving state-of-the-art per- formance on image classification with significantly less trainable parameters, its most light-weight version EfficientNet-B0 also suc- ceeded in benchmarking audio classification tasks [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We adopt EfficientNet-B0 as our encoder f W, resulting in 4M learnable pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We append a linear dense layer of J = dim θ neurons and a 1D batch normalization before tanh activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The goal of batch normalization is to gaussianize the input, such that the activated output is capable of uniformly cover the normalized prediction range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The input to f W is the log-scaled CQT coefficients of each example, computed with a filterbank spanning 10 octaves with 12 filters per octave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Physical: Functional transformation method (FTM) We are interested in the perpendicular displacement X(t, u) on a rectangular drum face, which can be solved from the following partial differential equation defined in the Cartesian coordinate system u = (u1, u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' �∂2X ∂t2 (t, u) − c2∇2X(t, u) � + S4� ∇4X(t, u) � + ∂ ∂t � d1X(t, u) + d3∇2X(t, u) � = Y(t, u) (10) In addition to the standard traveling wave equation in the first above parenthesis, the fourth-order spatial and first-order time derivatives incorporate damping factors induced by stiffness, internal friction in the drum material and air friction in the external environment, rendering the solution a closer simulation to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Specifically, α, S, c, d1, d3 designate respectively the side length ratio, stiffness, traveling wave speed, frequency-independent damping and frequency- dependent damping of the drum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Even though real world drums are mostly circular, a rectangular drum model is equally capable of eliciting representative percussive sounds in real world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The circular drum model simply requires a conversion of Equation 10 into the Polar coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We bound the four sides of this l by lα rectangular drum at zero at all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Moreover, we assume its excitation function to be separable and localized in space and time Y(t, u) = yu(u)δ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We implement generator g as a PDE solver to this high-order damped wave equation, namely the functional transformation method (FTM) [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' FTM solves the PDE by transforming the equation into its Laplace and functional space domain, where an algebraic solution can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' It then finds the time-space domain solution via inverse functional transforms, expressed in an infinite modal summation form x(t) = X(t, u) = � m∈N2 Km(u, t) exp(σmt) sin(ωmt) (11) The coefficients Km(u, t), σm, ωm are derived from the original PDE parameters in the following ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' ω2 m = (S4 − d2 3 4 )Γ2 m1,m2 + (c2 + d1d3 2 )Γm1,m2 − d2 1 4 (12) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Distributions of the sorted eigenvalues of M(θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' For the sake of comparison between PNP and P-loss, the dashed line indicates the eigenvalues of the identity matrix (see Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' σm = d3 2 Γm1,m2 − d1 2 (13) Km(u, t) = ym u δ(t) sin(πm1u1 l ) sin �πm2u2 lα � (14) where Γm1,m2 = π2m2 1/l2 + π2m2 2/(lα)2, and ym u is the mth coef- ficient associated to the eigenfunction sin(πmu/l) that decomposes yu(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Without losing connections to the acoustical manufacturing of the drum yet better relating g’s input with perceptual dimensions, we reparametrize the PDE parameters {S, c, d1, d3, α} into θ = {log ω1, τ1, log p, log D, α}, detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='4 of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We pre- scribe sonically-plausible ranges for each parameter in θ, normalize them between −1 and 1, uniformly sample in the hyper-dimensional cube, and obtain a dataset of 100k percussive sounds sampled at 22050 HZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The train/test/validation split is 8 : 1 : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' In particular, fundamental frequency ω1, duration τ1 falls into ranges [40, 1000] Hz and [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='4, 3] seconds respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Inhomoge- neous damping rate p, frequential dispersion D and aspect ratio α ranges are [10−5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='2], [10−5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='3], and [10−5, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Baselines We train fW with 3 different losses - multi-scale spectral loss [19], parameter loss, and PNP loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We use a batch size of 64 samples for spectral loss, and 256 samples for parameter and PNP loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The training proceeds for 70 epochs, where around 20% of the training set is seen at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We use Adam optimizer with learning rate 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Table 1 reports the training time per epoch on a single Tesla V100 16GB GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Evaluation with JTFS-based spectral loss We propose to use the L2 norm of JTFS coefficients error averaged over test set for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' As a point of reference, we also include the average multi-scale spectral error, implemented as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' One of the key distinctions between Euclidean JTFS distance and multi-scale spectral error is the former’s inclusion of spectro- temporal modulations information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Meanwhile unlike mean squared parameter error, both metrics reflect the perceptual closeness instead of parametric retrieval accuracy for each proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Discussion Despite being the optimal quadratic approximation of spectral loss, it is nontrivial to apply the bear PNP loss form as Equation 5 in OCD 15 10 5 0 5 log1o(on,j)Pitch JTFS distance (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' on test set) MSS (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' on test set) Training time per epoch P-loss Known 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='23 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='013 49 minutes DDSP with MSS loss Known 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='335 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='005 54 minutes PNP with JTFS loss Known 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='335 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='005 49 minutes DDSP with JTFS loss — — — est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=', > 1 day P-loss Unknown 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='91 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='094 53 minutes DDSP with MSS loss Unknown 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='95 ± 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='307 59 minutes PNP with JTFS loss Unknown 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='21 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='019 49 minutes Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Report of average JTFS distance and MSS metrics evaluated on test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Six models are trained with two modalities: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' the inclusion of pitch retrieval i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='regressing θ = {τ, log p, log D, α} vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' θ = {log ω1, τ, log p, log D, α}, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' the choice of loss function: P-loss, MSS loss, or PNP loss with adaptive damping mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The best performing models with known and unknown pitch are P-loss and PNP loss respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Training with MSS loss is more time consuming than training with P-loss or PNP loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Training with differentiable JTFS loss is unrealistic in the interest of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' On one hand, Φ◦g potentially has undesirable property that exposes the Riemannian metric calculations to numeri- cal precision errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' On the other hand, extreme deformation of the optimization landscape may lead to the same numerical instability facing stochastic gradient descent with spectral loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We report on a few remedies that helped stabilize learning with PNP loss, and offer insights on future directions to take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' First and foremost, our preliminary experiments show that train- ing PNP loss without damping λ = 0 subjects to serious convergence issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Recalling Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='2, indeed our empirical Ms suffer from high condition numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 2 shows the sorted eigenvalue distribu- tion of all Ms in test set, where Ms are rank-2,3 or 4 matrices with eigenvalues ranging from 0 to 1020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' This could be an implication that entries of θ contain implicit linear dependencies in generator g, or that local variations of certain θ fail to linearize differences in the output of g or Φ ◦ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' As an example, the aspect ratio α influences the modal frequencies and decay rates via [18, Equations 12–13], where in fact its variant 1/α + 1/α2 could be a better choice of variable that linearizes g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' To address Ms’ ill conditions we attempted at numerous damping mechanisms to update λ, namely constant λ, scheduled λ decay, and adaptive λ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The intuition is to have LPNP θ start in the parameter loss regime and move towards the spectral loss regime while training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The best performing model is achieved with adaptive λ decay, as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We initialize λ to be 1020 to match the largest empirical σ2 j , which later gets adaptively decayed to 3 × 1014 in 20 epochs, breaking records 7 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' This indicates that f W is able to learn with damped PNP loss, under the condition that λ being large enough to simulate parameter loss regime and compensate for deficiency in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The diagonal elements of M(θ) can be regarded as both the ap- plied weights’ magnitudes and proxies for θ’s perceptual significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' To gain further insights on how each model regresses different pa- rameters, we visualize in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='3 pairs of (|˜θ − θ|2 j, M(θ)j,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Three trends can be observed: First, τ and ω are regressed with the best accuracy across all learning objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Second, spectral loss particu- larly struggles in pitch ω and inharmonicity p retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Third, we may interpret x-axis as describing from left to right samples with increasing perceptual significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We observe that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='3(b), PNP loss is able to suppress more errors in samples with high M(θ)j,j than parameter loss, by a nonnegligible margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We believe that more of PNP loss’ mathematical potential can be exploited in the future, notably its ability to interpolate between various loss regimes and its use in hybrid optimization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' To start with, we plan to resort to a simpler differentiable synthesizer g Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' X-axis: weight assigned by PNP to one of the physical parameters in θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Y-axis: log squared estimation error for that same parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' α is omitted due to its poor retrieval results from all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' that guarantees a well-conditioned Riemannian metric M(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' More- over, we plan to explore other damping schemes and optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The current update mechanism, originated from the Leverberg-Marquardt Algorithm, aims to improve the conditioning of a matrix inversion problem in the Gauss-Newton algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' However when used jointly with stochastic gradient descent, each λ update may change the opti- mization landscape drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' The resulting optimization behavior is thus not fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We consider interfacing nonlinear least squares solver with SGD and forming a hybrid learning scheme in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' CONCLUSION In this article we have presented Perceptual-Neural-Physical (PNP) autoencoding, a bilinear form learning objective for sound matching task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' In our application, PNP optimizes the retrieval of physical parameters from sounds in a perceptually-motivated metric space, enabled by differentiable implementations of domain knowledge in physical modeling and computational proxy of neurophysiological construct of human auditory system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We demonstrated PNP’s mathematical proximity to spectral loss and its generalizability to parameter loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Using this formulation, we motivated and established one way of enabling smooth transition between optimizing in parameter and spectral loss regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' We have presented damping mechanisms to facilitate its learning under ill-conditioned empirical settings and discussed its mathematical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' w - w]2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' M[0,0] T|2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' M[1, 1] 100 10~2 10-3 10~5 Ploss 10~6 10-8 Spec 109 PNP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='0 0 100000 200000 300000 400000 (a) le10 (b) Ip -p]2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' M[2,2] ID - D2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' M[3,3] 101 100 10~3 10~2 10~6, 10-5 10-9 10-8 0 50000 100000 150000 200000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content='6 (c) (d) le106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' REFERENCES [1] Andrew Horner, “Wavetable matching synthesis of dynamic instruments with genetic algorithms,” Journal of the Audio Engineering Society, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 43, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 916–931, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [2] Jordie Shier, Kirk McNally, George Tzanetakis, and Ky Grace Brooks, “Manifold learning methods for visualization and browsing of drum machine samples,” Journal of the Audio Engineering Society, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 1/2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 40–53, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [3] Philippe Esling, Naotake Masuda, Adrien Bardet, Romeo De- spres, Axel Chemla, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=', “Universal audio synthesizer control with normalizing flows,” in Proceedings of the International Conference on Digital Audio Effects (DAFX), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [4] Leonardo Gabrielli, Stefano Tomassetti, Carlo Zinato, and Francesco Piazza, “End-to-end learning for physics-based acoustic modeling,” IEEE Transactions on Emerging Topics in Computational Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 160–170, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [5] Naotake Masuda and Daisuke Saito, “Synthesizer sound match- ing with differentiable DSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=',” in Proceedings of the Interna- tional Society on Music Information Retrieval (ISMIR) Confer- ence, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 428–434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [6] Martin Roth and Matthew Yee-king, “A comparison of para- metric optimization techniques for musical instrument tone matching,” Journal of the Audio Engineering Society, May 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [7] Matthew Yee-King, Leon Fedden, and Mark d’Inverno, “Au- tomatic programming of vst sound synthesizers using deep networks and other techniques,” IEEE Transactions on Emerg- ing Topics in Computational Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 150–159, 04 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [8] Jesse Engel, Lamtharn (Hanoi) Hantrakul, Chenjie Gu, and Adam Roberts, “DDSP: Differentiable Digital Signal Process- ing,” in Proceedings of the International Conference on Learn- ing Representations (ICLR), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [9] Joakim And´en, Vincent Lostanlen, and St´ephane Mallat, “Joint time–frequency scattering,” IEEE Transactions on Signal Pro- cessing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 3704–3718, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [10] Taishih Chi, Powen Ru, and Shihab A Shamma, “Multiresolu- tion spectrotemporal analysis of complex sounds,” The Journal of the Acoustical Society of America, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 118, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 887– 906, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [11] Vincent Lostanlen, Christian El-Hajj, Mathias Rossignol, Gr´egoire Lafay, Joakim And´en, and Mathieu Lagrange, “Time– frequency scattering accurately models auditory similarities between instrumental playing techniques,” EURASIP Journal on Audio, Speech, and Music Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 2021, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 1–21, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [12] Mathieu Andreux, Tom´as Angles, Georgios Exarchakis, Roberto Leonarduzzi, Gaspar Rochette, Louis Thiry, John Zarka, St´ephane Mallat, Joakim And´en, Eugene Belilovsky, Joan Bruna, Vincent Lostanlen, Muawiz Chaudhary, Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Hirn, Edouard Oyallon, Sixin Zhang, Carmine Cella, and Michael Eickenberg, “Kymatio: Scattering transforms in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=',” Journal of Machine Learning Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 60, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 1–6, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [13] John Muradeli, Cyrus Vahidi, Changhong Wang, Han Han, Vin- cent Lostanlen, Mathieu Lagrange, and George Fazekas, “Dif- ferentiable time-frequency scattering in kymatio,” in Proceed- ings of the International Conference on Digital Audio Effects (DAFX), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [14] Mingxing Tan and Quoc Le, “EfficientNet: Rethinking model scaling for convolutional neural networks,” in Proceedings of the International conference on Machine Learning (ICML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' PMLR, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 6105–6114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [15] Neil Zeghidour, Olivier Teboul, F´elix de Chaumont Quitry, and Marco Tagliasacchi, “Leaf: A learnable frontend for audio classification,” ICLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Trautmann and Rudolf Rabenstein, Digital Sound Synthesis by Physical Modeling Using the Functional Transformation Method, 01 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [17] Maximilian Sch¨afer, Manuel Werner, and Rudolf Rabenstein, “Physical modeling in sound synthesis: Vibrating plates,” 05 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [18] Han Han and Vincent Lostanlen, “wav2shape: Hearing the Shape of a Drum Machine,” in Proceedings of Forum Acusticum, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' 647–654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' [19] Christian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Steinmetz and Joshua D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} +page_content=' Reiss, “auraloss: Audio focused loss functions in PyTorch,” in Digital Music Research Network One-day Workshop (DMRN+15), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfEAP6/content/2301.02886v1.pdf'} diff --git a/DdE3T4oBgHgl3EQfUwqw/content/tmp_files/2301.04454v1.pdf.txt b/DdE3T4oBgHgl3EQfUwqw/content/tmp_files/2301.04454v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9e7f9d3d1fb8cb432ad6f884866aff9ec68022c --- /dev/null +++ b/DdE3T4oBgHgl3EQfUwqw/content/tmp_files/2301.04454v1.pdf.txt @@ -0,0 +1,593 @@ +Allo-centric Occupancy Grid Prediction for Urban Traffic Scene Using +Video Prediction Networks +Rabbia Asghar1, Lukas Rummelhard1, Anne Spalanzani1, Christian Laugier1 +Abstract— Prediction of dynamic environment is crucial to +safe navigation of an autonomous vehicle. Urban traffic scenes +are particularly challenging to forecast due to complex interac- +tions between various dynamic agents, such as vehicles and +vulnerable road users. Previous approaches have used ego- +centric occupancy grid maps to represent and predict dynamic +environments. However, these predictions suffer from blurri- +ness, loss of scene structure at turns, and vanishing of agents +over longer prediction horizon. In this work, we propose a +novel framework to make long-term predictions by representing +the traffic scene in a fixed frame, referred as allo-centric +occupancy grid. This allows for the static scene to remain fixed +and to represent motion of the ego-vehicle on the grid like +other agents’. We study the allo-centric grid prediction with +different video prediction networks and validate the approach +on the real-world Nuscenes dataset. The results demonstrate +that the allo-centric grid representation significantly improves +scene prediction, in comparison to the conventional ego-centric +grid approach. +Index Terms— Scene Prediction, Deep Learning, Autonomous +Vehicles +I. INTRODUCTION +Prediction of traffic scene evolution is essential to an +autonomous vehicle for planning as well as detecting dan- +gerous situations. In urban traffic scenarios, the vehicles not +only interact with other vehicles, but also share space with +vulnerable road users such as pedestrians and cyclists. Key +challenges involve the uncertainty and multi-modality of the +behaviour of agents in the environment, and complex multi- +agents interactions [1]. While human drivers show superior +ability to forecast the agents’ behaviour and interactions in +such traffic scenes, it remains a challenge for autonomous +vehicles. +Data-driven methods provide powerful tools to solve pre- +diction problems, particularly dealing with complex social +interactions [2]. Most conventional approaches are object +or agent-based and rely on heavily pre-processed data [3], +[4]. Dynamic Occupancy Grip Maps (DOGMs), on the other +hand, allow for end-to-end learning due to their discretized +spatial representation, without higher-level segmentation [5]. +Additionally, DOGMs are versatile in terms of sensor depen- +dency, and can be generated from a variety of raw sensor +data, such as Lidar or camera images. In our work, we +use Bayesian-filter-based DOGM [6] that provide us with +a spatially-dense model representation of static and dynamic +space, as well as free and unknown space in the environment, +as shown in Fig1. +1 Univ. Grenoble Alpes, Inria, 38000 Grenoble, France, email: First- +Name.LastName@inria.fr +As the DOGM is generated using data from the vehicle- +mounted sensors, the grid is traditionally ego-centric,i.e. the +position of ego-vehicle is fixed in the grid. While this is an +effective method in scene representation, it complicates the +long-term prediction problem. For a dynamic ego-vehicle, +the complete scene translates and/or rotates around the ego- +vehicle, even the static components in the scene. Therefore, +the prediction network must transform every cell in the +grid, leading to blurry and vanishing static scene at longer +prediction time horizons. +To address this, we instead generate DOGMs with respect +to a fixed reference frame, referred as allo-centric grid. +While the observed space around the ego-vehicle remains +the same, the static scene structure in the allo-centric grid +remains fixed. This is illustrated in Fig. 1 where the ego- +vehicle is encircled, the vehicle moves like other agents in +the scene. +We approach the long-term multi-step predictions of allo- +centric DOGM as a video prediction problem due to the +inherent similarities between an image and an occupancy +grid, and both being a spatio-temporal problem [7]. Results +incorporating different video prediction networks are stud- +ied, including state-of-the-art recurrent neural networks and +memory-augmented network approaches. We compare and +evaluate the prediction results of allo-centric and ego-centric +grids for identical scenes and demonstrate the superior per- +formances of the allo-centric grid predictions. +The proposed approach is validated with the real-world +NuScenes dataset [3] of urban traffic scenes. We show that +allo-centric grids significantly improve the prediction results +and demonstrate the ability to retain the scene structure and +learn behaviours. +The paper is organized as follows. Section II discusses +related work to video and scene predictions. Section III +describes the system overview. Section IV and V present +implementations, results and analysis. Finally conclusions +are drawn in section VI. +II. RELATED WORK +A. Video Prediction +Spatio-temporal deep-learning methods have been ef- +fectively used for video prediction problems. Commonly, +combinations of Convolutional Neural Networks (CNNs) +and Recurrent Neural Networks (RNNs) are incorporated. +CNNs are capable of extracting spatial information and +capturing inter-dependencies of the surrounding pixels while +RNNs, such as long short-term memory (LSTM) blocks, +arXiv:2301.04454v1 [cs.CV] 11 Jan 2023 + +Fig. 1: Overview of our proposed approach. The allo-centric DOGM is represented as an image. Each channel red, green and +blue represent unknown, dynamic and static cells respectively. The black space represents known free space. The ego-vehicle +is circled in dotted line in both input and target output sequences. +capture the sequential or temporal dependencies. Lotter et +al. proposed Predictive Coding Network (PredNet), a deep +learning network architecture that comprises of vertically- +stacked Convolutional LSTMs (ConvLSTMs) where the local +error and the prediction signal are propagated bottom-up +and top-down respectively [8]. Wang et al. addresses the +video prediction challenges of capturing short-term and long- +term dynamics with the PredRNN architecture [9]. Building +on their original approach [10], they introduce memory- +decoupled spatio-temporal LSTM (ST-LSTM) blocks, fea- +ture zigzag memory flow and a novel curriculum learning +strategy to improve prediction results. Kim et al. takes in- +spiration from memory-augmented networks to use external +memory (LMC-Memory) to learn and store long-term motion +dynamics and propose a memory query decomposition to ad- +dress the high-dimensionality of motions in video predictions +[11]. +B. Occupancy Grid Prediction +Jeon et al. proposed conventional ConvLSTM to predict +interaction-intensive traffic scenes on occupancy grids [12]. +The approach represents only vehicles in the occupancy grid, +their states extracted from camera inputs. Desquaire et al. +[13], proposed an end-to-end object-tracking approach by +incorporating directly Lidar sensor data to predict the binary +grid, using recurrent neural network. To incorporate ego- +vehicle motion, they utilize a spatial transformer to allow +internal memory of RNNs to learn environment of the state. +Mohajerin et al. [14] suggested an RNN-based architecture +with a difference learning method, and makes OGM pre- +diction in the field of view of ego-vehicle front camera. +Schreiber et al. [15] proposed an encoder-decoder network +architecture, along with skip connections, to make long-term +DOGM predictions. While they collect the sensor data from +an autonomous vehicle, the vehicle remains stationary and +only acts as the sensor collection point at different inter- +sections. Itkina et al. proposed to use evidential occupancy +grid and implement PredNet architecture for the prediction +[16]. The approach is then carried forward to develop the +double-pronged architecture [17] and attention-augmented +ConvLSTM [18]. The latter work is able to make long-term +predictions, however at turns the predictions still lose the +scene structure. Mann et al. [19] addressed the problem of +OGM prediction in urban scenes by incorporating vehicles +semantics in the environment. Their proposed method de- +pends on the annotated vehicle data labels available in the +dataset. +Contrary to the conventional Occupancy Grid Prediction, +we present an allo-centric DOGM representation to predict +the urban traffic scene with respect to a fixed reference frame. +Apart from the conventional recurrent representation learning +approaches, we also use memory-augmented learning-based +video-prediction method, in relevance to learning long-term +motion context of the dynamic agents. +III. SYSTEM OVERVIEW +We discuss here the overall proposed approach for allo- +centric DOGM prediction, the pipeline is summarized in Fig. +1. +A. Dynamic Occupancy Grid Maps +Dynamic occupancy grid maps provide a discretized rep- +resentation of environment in a bird’s eye view, where every +cell in the grid is independent and carries information about +the associated occupancy and velocity. +To generate DOGMs, we incorporate the Conditional +Monte Carlo Dense Occupancy Tracker (CMCDOT) [6]. +This approach associates four occupancy states to the grid. +Each cell carries the probabilities of the cell being i) occupied +and static, ii) occupied and dynamic, iii) unoccupied or free +and iv) if the occupancy is unknown. The probabilities of +these four states sum to one. In our work, we make use +of three of these states and represent the grid as an RGB +image. The channels Red, Green and Blue represent the +unknown state, dynamic state and static state respectively. +The associated probabilities of the cell in the 3-channel +DOGM grid are interpreted as the pixel values of the RGB +images. The RGB grid images can be seen in Fig. 1-2. Low +probabilities in all three channels leave the grid-image black, +therefore, representing free space. +For allo-centric grid generation, we define the grid in the +world frame, close to the initial position of ego-vehicle. +The state probabilities are initially computed in an ego- +centric grid, since we use the on-board sensor data. To ensure +that we have cell information for the complete allo-centric +grid dimensions when the vehicle is dynamic and moving +away from the world frame origin, a much larger ego-centric + +Allo-centric +Video +DOGM +Prediction +Generation +NetworkDOGM is computed. This information is then fused to update +every cell states in the allo-centric grid in the world frame. +We compare the allo-centric and ego-centric grids at 4 +time instants for the same scene and same grid dimensions in +Figure 2. In the allo-centric grid, the ego-vehicle (illustrated +in the pink box) can be seen moving with respect to the grid, +while it remains fixed in the ego-centric grid. It is important +to note that the observable space around the ego-vehicle +remains the same for both grids. However, since they are +defined in different frames, the two cover different spaces in +the scene at a given time. We illustrate the common space +covered by both grids since the start of the sequence, marked +by yellow boundary. +Fig. 2: Visualization of allo-centric and ego-centric grids, +generated for the same scene. The area marked by yellow +lines is the common region covered by both grids up until +the t-th sequence. The ego-vehicle is boxed in pink grid and +the bus passing by is encircled in white. +B. Problem Formulation +We formally define the task of predicting the scene in +allo-centric DOGM representation, as sequence-to-sequence +learning, see Fig. 1. A sequence comprises of a set of +sequential grid images that capture the evolution of a given +scene. Let Xt ∈ R3xW xH and Yt ∈ R3xW xH be the t-th frame +of the 3-channel grid-image where W and H denote the width +and height respectively. The input sequence for the grid- +image is denoted by Xt−N:t, representing N consecutive +frames. Given a set of input sequence, the task of the +network is to predict future grid images, i.e. output sequence. +The target and predicted output sequences are denoted by +Yt+1:t+P and ˆYt+1:t+P where P is the prediction horizon. +For training and testing data, the DOGMs can be generated +for both the input and the target sequences, leaving behind +no additional need for labelled data or human intervention. +Since the input sequences, Xt−N:t, and output sequences, +Yt+1:t+P , are represented as images, this prediction task can +be considered a video prediction problem. +C. Deep Learning Prediction Architectures +To study and compare the scene prediction with ego- +centric and allo-centric grids, we train our datasets with +different video prediction networks. We consider 3 networks, +briefly discussed in section II-A: PredNet, PredRNN, LMC- +Memory with memory alignment learning (here on referred +as LMC-Memory). +PredNet [8], inspired from predictive coding, makes pre- +dictions based on how the predicted frames deviate from the +target [20]. The original work tests the network on vehicle +mounted camera images from Kitti dataset [21] and demon- +strates the ability to capture both egocentric motion as well as +motion of objects in camera images. We consider PredRNN +[9] and LMC-Memory architecture [11] as the state of the +art video prediction networks that aim to capture long-term +dependencies and motion context. PredRNN implements +novel ST-LSTM units with a zigzag internal memory flow +and proposes memory decoupling loss to discourage learning +redundant features. LMC-Memory architecture, on the other +hand, proposes an external memory block with its own +parameters to store various motion contexts. The approach +also offers an efficient computation method since the motion +context for long-term multi-step predictions is computed only +once for a given input sequence. +We study these networks capabilities to retain the occu- +pancy of the static region, and the ability to predict motion +of dynamic agents in DOGM. +D. Unknown Channel and Loss functions +In both ego-centric and allo-centric grids, a significant +part of the scene remains unobserved, see Fig. 2 (unknown +channel is represented in red). This is more pronounced in +the initial frames of the allo-centric grid, where the Lidar is +unable to detect the farthest area from the ego-vehicle. +While it is more relevant to learn the evolution of static +and dynamic components in the scene, inclusion of unknown +channel is useful for our prediction task. A Lidar based grid +is often unable to capture the full shape of a vehicle. For +example, we can see in Fig. 2 how the occupied cells by the +bus vary in different time steps on the grid. It is only in the +2.0s time step that a rectangular shape is observed, otherwise +different parts of the bus remain unknown. The unknown +channel at different instants also carries spatial information +of the agents with respect to the ego-vehicle. Thus, with the +sequential frames and the unknown channel, we assist the +network to be able to extract spatial information and learn +scene representation. +The inclusion of unknown channel and emphasis on +learning static and dynamic components is addressed in the +loss function. Loss function L in the implemented video +prediction networks is modified to carry the weighted sum +of the RGB channels: +L = αLR + β(LG + LB) +(1) +where, +LR, LG and LB represent the loss for unknown (red), +dynamic (green) and static channels (blue) respectively. In +order to encourage the network to learn and improve the +prediction of the static and dynamic channels, α is always +kept smaller than β. + +IV. EXPERIMENTS +A. Dataset +We study the prediction performance on the real-world +NuScenes dataset [3]. The original dataset consists of 850 +scenes for training and 150 scenes for testing, each scene +is approximately 20s long. We generate the DOGM grid- +based on the Lidar pointcloud and available odometry. For +allo-centric grid, we represent the scene with respect to a +fixed reference frame and a grid dimension of 60 x 60m, +with a resolution of 0.1m per cell. Each sequence starts with +the ego-vehicle heading facing up, capturing the scene 10m +behind and 50m ahead of it. The initial pose was selected to +ensure that the ego-vehicle remains within the grid for the +total sequence length, even when running at a high speed. For +egocentric grid, we generate a grid of the same dimensions +and resolution, and the ego-vehicle fixed in the center. Each +sequence is comprised of 35 frames, a time duration of 3.5s +with DOGM grid images generated every 0.1s. In total, we +have 4,250 training and 750 testing sequences respectively. +B. Training +The input sequence Xt−9:t consists of 10 frames (1.0s). +Each network is trained to make predictions ˆYt+1:t+25 for +25 future frames (2.5s). Both the allo-centric and ego- +centric datasets are trained with the original parameters of +the respective video prediction network. For training with +PredRNN and LMC Memory networks, both allo-centric +and ego-centric grid images are resized to 192x192 pixels. +PredRNN is trained with a batch size of 4 and a learning +rate of 10−4. The number of channels of each hidden state is +set to 64. The loss function is the sum of L2 and decoupling +loss, and the values of α and β in Eq. (1) are set to 0.2 and +0.8. LMC-Memory is trained with a learning rate of 2x10−4, +memory slot is set to 100 and ConvLSTM to 4 layers for +frame predictions. The loss function is the sum of L1 and +L2 losses. The values of α and β are set to 0.2 and 0.8. +For training with PredNet, the grid images are resized to +160x160 pixels. The network is set to 4 hierarchical layers +with an initial learning rate of 10−3. The loss function is the +L1 loss of only the first layer, the values of α and β are set +to 0.05 and 0.8. All models are trained on Adam optimizer +for 30 epochs. +V. EVALUATION +For evaluation, we are particularly interested in static and +dynamic agents in the scene. We discussed in section III-D, +the utility of unknown regions in learning scene representa- +tion. But the unknown region occupies a big portion of the +grid and, thus, in evaluation, overshadows the performance of +more interesting and relevant segments: static and dynamic +regions. For this reason, we evaluate the dataset and network +performances based on two channels of the predicted images, +the blue and green channels representing static and dynamic +components in the scene. We encourage the readers to refer +to the video1 for a better visualization of the results. +1https://youtu.be/z-0BVM93X8c +A. Quantitative Evaluation +The allo-centric and ego-centric grids at any instant ob- +serve different parts of the scene, see Fig. 2. For fair +comparison between them, we modify the test dataset and +crop out the part of each t-frame that has not been observed +until the t-th sequence by both grids. Thus, for example, the +part of the grids outside of the yellow dotted boxes in Fig. +2 are blacked out for the input sequence frames Xt−N:t as +well as the target frames in the output sequence Yt+1:t+P . +We measure the performances using three metrics: MSE +(Mean Square Error), SSIM (Structured Similarity Indexing +Method), and LPIPS (Learned Perceptual Image Patch Sim- +ilarity) [22]. MSE is calculated by the pixel-wise difference +between the ground truth and the predicted frame per channel +and per cell. However, with MSE, the slightest error in +predicted motion can result in large errors in the ego- +centric grids dataset. The SSIM and LPIPS metrics evaluate +the prediction results based on the structural similarity and +perception similarity respectively. Lower values are better for +MSE and LPIPS while higher values are better for SSIM. +Table I shows average results for the complete 2.5s pre- +diction horizons. The MSE score of allo-centric grids is +significantly lower compared to the one of ego-centric grids. +Since the complete scene transforms with respect to the +ego-vehicle, the MSE is always higher in the ego-centric +grid. The SSIM and LPIPS scores are also significantly +superior for the allo-centric grid, due to the tendency of ego- +centric grids to get increasingly blurry for higher prediction +horizons. +Network +MSE x 10−2(↓) +SSIM(↑) +LPIPS(↓) +Allo-centric grid +LMC-Memory +0.894 +0.895 +0.167 +PredRNN +0.882 +0.904 +0.167 +PredNet +0.905 +0.888 +0.172 +Ego-centric grid +LMC-Memory +1.302 +0.856 +0.217 +PredRNN +1.138 +0.845 +0.234 +PredNet +1.335 +0.847 +0.225 +TABLE I: Average results with allo-centric and ego-centric +grids for prediction horizon of 2.5s. The allocentric grid +outperforms the other in all three video prediction networks. +In Fig. 3, we plot scores of the metrics for every 0.5s +prediction step. The results with allo-centric grid (shown +in blue) always perform better than the ego-centric grids. +Among the three prediction networks, overall PredRNN +performs the best with allo-centric grids. However, with the +ego-centric grids (results shown in orange), PredRNN offers +a good MSE score but the SSIM and LPIPS performances +drop after 1.0s. This is because PredRNN tends to make +blurry and diffused predictions in the output frames; this +helps reduce the MSE but the scene loses its structures. This +is further seen in the qualitative results discussed in section +V-B and illustrated in Fig. 4. +B. Qualitative Evaluation +The prediction results between the allo-centric and ego- +centric grids differ drastically when the ego-vehicle is turning + +Fig. 3: Results with the MSE(↓), SSIM(↑) and LPIPS(↓) metrics with allo-centric and ego-centric grids for input sequences +of 1.0s and prediction horizon up to 2.5s. For fair comparison, all test sequence frames were modified to only contain the +scene observable in both the allo-centric and ego-centric grids. The allo-centric grid (results plotted in blue) outperforms +the other with all three video prediction networks. +(a) Allo-centric grids +(b) Ego-centric grids +Fig. 4: Qualitative results for the ego-vehicle leaving a roundabout on both allo-centric (4a) and ego-centric grids (4b). The +input sequence consists of 10 frames (1.0s) and output predicted sequence of up to 25 frames (2.5s). The prediction results +are shown at 0.5s, 1.5s and 2.5s instants and are magnified at the interesting spaces, marked by red box in the target(ground +truth) frames. The best results can be observed with LMC-Memory network with the allo-centric grid that retains the scene +structure and predicts the motion of the ego-vehicle best. +at an intersection or driving on a curved road. Figure 4 shows +results for a sequence where the ego-vehicle is exiting a +roundabout. In this scene, while there are no other dynamic +agents, the network needs to predict the behaviour of the ego- +vehicle, when it is driving along the curved static segment +(alluding to road structure) and is headed towards static +objects/obstacles. For the allo-centric grids, the challenge is +to predict the ego-vehicle pose while the scene remains static. +The best results are achieved with the LMC-Memory. The +vehicle pose is well-predicted up to 2.5s, its orientation is +adjusted so that it does not hit the static components. For +the same grid, the PredRNN fails to learn and predict the +behaviour resulting in false prediction of collisions. The ego- +vehicle, while getting more blurry, diffuses into the static +obstacles on the road. With the PredNet, the ego vehicle +is almost already lost at 1.5s prediction horizon. This is +expected behaviour since PredNet is ideally not aimed at +long-term video predictions. With all three networks, the ego- +vehicle gets more blurry, however with PredRNN, the static +scene also tends to get blurry at larger prediction horizon. + +洋LMC-ego +PredRNN-ego +PredNet-ego +LMC-allo +PredRNN-allo +PredNet-alloIn the ego-centric grid, the whole scene rotates around the +ego-vehicle. LMC-memory and PredNet significantly lose +the static components ahead of the vehicle. The rotation +results in increasing blurriness at every time step. PredRNN +predictions are more diffused and faint blurry cells are +still visible ahead of the vehicle, even at 2.5s prediction +horizon. In context of planning and safe navigation, this +high uncertainty in the environment structure renders the +prediction results unreliable. +VI. DISCUSSION AND FUTURE WORK +In this work, we presented a novel allo-centric dynamic +ocuupancy grid approach for long-term prediction of urban +traffic scene, and compared it to the conventional ego- +centric DOGM approach. We trained and tested various +video prediction networks to show that allo-centric DOGM +representation has superior ability to predict the same scene. +The most significant improvement is the allo-centric grid’s +ability to retain the static scene structure, especially when the +vehicle turns. The ego-centric grid, on the other hand tends +to lose the static scene, and hence the crucial information +about whether the given space is occupied or free. +The results of allo-centric grids prediction with state-of- +the-art PredRNN and LMC-Memory approaches have shown +complementary benefits. PredRNN predictions, though dif- +fuse and get more blurry, are capable of maintaining agents +longer. We observe that LMC-memory shows better tendency +at learning behaviours in comparison to the PredRNN. +It is pertinent to mention here that the two grids are still +very similar. In both scenarios, the observable space updates +relative to the position of the vehicle in the scene. Thus, +in allo-centric grid while the grid is no more fixed to the +ego-vehicle, the ego-vehicle bias remains. +All three video prediction networks tested in this work +address the prediction problem as deterministic. However, +the behaviour of agents in urban traffic scene tends to be +multimodal. For future work, the addition of multimodal +prediction capabilities in the network architecture would be +interesting. Additionally, the incorporation of semantics in +the occupancy grid such as agent type and offline road infor- +mation could assist in learning behaviours and interactions. +REFERENCES +[1] S. Mozaffari, O. Y. Al-Jarrah, M. Dianati, P. Jennings, and A. Mouza- +kitis, “Deep learning-based vehicle behavior prediction for au- +tonomous driving applications: A review,” IEEE Transactions on +Intelligent Transportation Systems, vol. 23, no. 1, pp. 33–47, 2020. +[2] A. Alahi, K. Goel, V. Ramanathan, A. Robicquet, L. Fei-Fei, and +S. Savarese, “Social lstm: Human trajectory prediction in crowded +spaces,” in Proceedings of the IEEE conference on computer vision +and pattern recognition, 2016, pp. 961–971. +[3] H. Caesar, V. Bankiti, A. H. Lang, S. Vora, V. E. Liong, Q. Xu, A. Kr- +ishnan, Y. Pan, G. Baldan, and O. Beijbom, “nuscenes: A multimodal +dataset for autonomous driving,” arXiv preprint arXiv:1903.11027, +2019. +[4] S. Ettinger, S. Cheng, B. Caine, C. Liu, H. Zhao, S. Pradhan, +Y. Chai, B. Sapp, C. R. Qi, Y. Zhou et al., “Large scale interactive +motion forecasting for autonomous driving: The waymo open motion +dataset,” in Proceedings of the IEEE/CVF International Conference +on Computer Vision, 2021, pp. 9710–9719. +[5] A. N`egre, L. Rummelhard, and C. Laugier, “Hybrid sampling bayesian +occupancy filter,” in 2014 IEEE Intelligent Vehicles Symposium Pro- +ceedings. +IEEE, 2014, pp. 1307–1312. +[6] L. Rummelhard, A. N`egre, and C. Laugier, “Conditional monte carlo +dense occupancy tracker,” in 2015 IEEE 18th International Conference +on Intelligent Transportation Systems. +IEEE, 2015, pp. 2485–2490. +[7] S. Oprea, P. Martinez-Gonzalez, A. Garcia-Garcia, J. A. Castro- +Vargas, S. Orts-Escolano, J. Garcia-Rodriguez, and A. Argyros, “A +review on deep learning techniques for video prediction,” IEEE +Transactions on Pattern Analysis and Machine Intelligence, 2020. +[Online]. Available: https://arxiv.org/pdf/2004.05214.pdf +[8] W. Lotter, G. Kreiman, and D. Cox, “Deep predictive coding networks +for video prediction and unsupervised learning,” 5th International +Conference on Learning Representations, ICLR 2017 - Conference +Track Proceedings, pp. 1–18, 2017. +[9] Y. Wang, H. Wu, J. Zhang, Z. Gao, J. Wang, P. Yu, and M. Long, +“Predrnn: A recurrent neural network for spatiotemporal predictive +learning,” IEEE Transactions on Pattern Analysis and Machine Intel- +ligence, pp. 1–1, 2022. +[10] Y. Wang, M. Long, J. Wang, Z. Gao, and P. S. Yu, “Predrnn: Recurrent +neural networks for predictive learning using spatiotemporal lstms,” in +Advances in Neural Information Processing Systems, I. Guyon, U. V. +Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and +R. Garnett, Eds., vol. 30. +Curran Associates, Inc., 2017. +[11] S. Lee, H. G. Kim, D. H. Choi, H.-I. Kim, and Y. M. Ro, “Video +prediction recalling long-term motion context via memory alignment +learning,” in Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, 2021, pp. 3054–3063. +[12] H.-S. Jeon, D.-S. Kum, and W.-Y. Jeong, “Traffic Scene Prediction +via Deep Learning: Introduction of Multi-Channel Occupancy Grid +Map as a Scene Representation,” in 2018 IEEE Intelligent Vehicles +Symposium (IV), 2018, pp. 1496–1501. +[13] J. Dequaire, P. Ondr´uˇska, D. Rao, D. Wang, and I. Posner, +“Deep tracking in the wild: End-to-end tracking using recurrent +neural networks,” The International Journal of Robotics Research, +vol. 37, no. 4-5, pp. 492–512, jun 2017. [Online]. Available: +https://doi.org/10.1177/0278364917710543 +[14] N. Mohajerin and M. Rohani, “Multi-step prediction of occupancy grid +maps with recurrent neural networks,” in Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recognition, 2019, pp. +10 600–10 608. +[15] M. Schreiber, S. Hoermann, and K. Dietmayer, “Long-term occupancy +grid prediction using recurrent neural networks,” in 2019 International +Conference on Robotics and Automation (ICRA). +IEEE, 2019, pp. +9299–9305. +[16] M. Itkina, K. Driggs-Campbell, and M. J. Kochenderfer, “Dynamic +environment prediction in urban scenes using recurrent representation +learning,” in 2019 IEEE Intelligent Transportation Systems Conference +(ITSC). +IEEE, 2019, pp. 2052–2059. [Online]. Available: https: +//arxiv.org/abs/1904.12374 +[17] M. Toyungyernsub, M. Itkina, R. Senanayake, and M. J. Kochenderfer, +“Double-Prong ConvLSTM for Spatiotemporal Occupancy Prediction +in Dynamic Environments,” arXiv preprint arXiv:2011.09045, 2020. +[Online]. Available: http://arxiv.org/abs/2011.09045 +[18] B. Lange, M. Itkina, and M. J. Kochenderfer, “Attention Aug- +mented ConvLSTM for Environment Prediction,” arXiv preprint +arXiv:2010.09662, 2020. +[19] K. S. Mann, A. Tomy, A. Paigwar, A. Renzaglia, and C. Laugier, +“Predicting future occupancy grids in dynamic environment with +spatio-temporal learning,” 2022. [Online]. Available: https://arxiv.org/ +abs/2205.03212 +[20] R. P. Rane, E. Sz¨ugyi, V. Saxena, A. Ofner, and S. Stober, “Prednet +and predictive coding: A critical review,” in Proceedings of the 2020 +international conference on multimedia retrieval, 2020, pp. 233–241. +[21] A. Geiger, P. Lenz, C. Stiller, and R. Urtasun, “Vision meets robotics: +The kitti dataset,” International Journal of Robotics Research (IJRR), +2013. +[22] R. Zhang, P. Isola, A. A. Efros, E. Shechtman, and O. Wang, “The +unreasonable effectiveness of deep features as a perceptual metric,” in +CVPR, 2018. + diff --git a/DdE3T4oBgHgl3EQfUwqw/content/tmp_files/load_file.txt b/DdE3T4oBgHgl3EQfUwqw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d111cad85500d2251ac3ac53dda0723f3409231 --- /dev/null +++ b/DdE3T4oBgHgl3EQfUwqw/content/tmp_files/load_file.txt @@ -0,0 +1,493 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf,len=492 +page_content='Allo-centric Occupancy Grid Prediction for Urban Traffic Scene Using Video Prediction Networks Rabbia Asghar1, Lukas Rummelhard1, Anne Spalanzani1, Christian Laugier1 Abstract— Prediction of dynamic environment is crucial to safe navigation of an autonomous vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Urban traffic scenes are particularly challenging to forecast due to complex interac- tions between various dynamic agents, such as vehicles and vulnerable road users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Previous approaches have used ego- centric occupancy grid maps to represent and predict dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' However, these predictions suffer from blurri- ness, loss of scene structure at turns, and vanishing of agents over longer prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In this work, we propose a novel framework to make long-term predictions by representing the traffic scene in a fixed frame, referred as allo-centric occupancy grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' This allows for the static scene to remain fixed and to represent motion of the ego-vehicle on the grid like other agents’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We study the allo-centric grid prediction with different video prediction networks and validate the approach on the real-world Nuscenes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The results demonstrate that the allo-centric grid representation significantly improves scene prediction, in comparison to the conventional ego-centric grid approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Index Terms— Scene Prediction, Deep Learning, Autonomous Vehicles I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' INTRODUCTION Prediction of traffic scene evolution is essential to an autonomous vehicle for planning as well as detecting dan- gerous situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In urban traffic scenarios, the vehicles not only interact with other vehicles, but also share space with vulnerable road users such as pedestrians and cyclists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Key challenges involve the uncertainty and multi-modality of the behaviour of agents in the environment, and complex multi- agents interactions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' While human drivers show superior ability to forecast the agents’ behaviour and interactions in such traffic scenes, it remains a challenge for autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Data-driven methods provide powerful tools to solve pre- diction problems, particularly dealing with complex social interactions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Most conventional approaches are object or agent-based and rely on heavily pre-processed data [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Dynamic Occupancy Grip Maps (DOGMs), on the other hand, allow for end-to-end learning due to their discretized spatial representation, without higher-level segmentation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Additionally, DOGMs are versatile in terms of sensor depen- dency, and can be generated from a variety of raw sensor data, such as Lidar or camera images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In our work, we use Bayesian-filter-based DOGM [6] that provide us with a spatially-dense model representation of static and dynamic space, as well as free and unknown space in the environment, as shown in Fig1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Grenoble Alpes, Inria, 38000 Grenoble, France, email: First- Name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='LastName@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='fr As the DOGM is generated using data from the vehicle- mounted sensors, the grid is traditionally ego-centric,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' the position of ego-vehicle is fixed in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' While this is an effective method in scene representation, it complicates the long-term prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For a dynamic ego-vehicle, the complete scene translates and/or rotates around the ego- vehicle, even the static components in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Therefore, the prediction network must transform every cell in the grid, leading to blurry and vanishing static scene at longer prediction time horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' To address this, we instead generate DOGMs with respect to a fixed reference frame, referred as allo-centric grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' While the observed space around the ego-vehicle remains the same, the static scene structure in the allo-centric grid remains fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1 where the ego- vehicle is encircled, the vehicle moves like other agents in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We approach the long-term multi-step predictions of allo- centric DOGM as a video prediction problem due to the inherent similarities between an image and an occupancy grid, and both being a spatio-temporal problem [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Results incorporating different video prediction networks are stud- ied, including state-of-the-art recurrent neural networks and memory-augmented network approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We compare and evaluate the prediction results of allo-centric and ego-centric grids for identical scenes and demonstrate the superior per- formances of the allo-centric grid predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The proposed approach is validated with the real-world NuScenes dataset [3] of urban traffic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We show that allo-centric grids significantly improve the prediction results and demonstrate the ability to retain the scene structure and learn behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Section II discusses related work to video and scene predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Section III describes the system overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Section IV and V present implementations, results and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Finally conclusions are drawn in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Video Prediction Spatio-temporal deep-learning methods have been ef- fectively used for video prediction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Commonly, combinations of Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs) are incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' CNNs are capable of extracting spatial information and capturing inter-dependencies of the surrounding pixels while RNNs, such as long short-term memory (LSTM) blocks, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='04454v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='CV] 11 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1: Overview of our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The allo-centric DOGM is represented as an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Each channel red, green and blue represent unknown, dynamic and static cells respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The black space represents known free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The ego-vehicle is circled in dotted line in both input and target output sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' capture the sequential or temporal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Lotter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' proposed Predictive Coding Network (PredNet), a deep learning network architecture that comprises of vertically- stacked Convolutional LSTMs (ConvLSTMs) where the local error and the prediction signal are propagated bottom-up and top-down respectively [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' addresses the video prediction challenges of capturing short-term and long- term dynamics with the PredRNN architecture [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Building on their original approach [10], they introduce memory- decoupled spatio-temporal LSTM (ST-LSTM) blocks, fea- ture zigzag memory flow and a novel curriculum learning strategy to improve prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' takes in- spiration from memory-augmented networks to use external memory (LMC-Memory) to learn and store long-term motion dynamics and propose a memory query decomposition to ad- dress the high-dimensionality of motions in video predictions [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Occupancy Grid Prediction Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' proposed conventional ConvLSTM to predict interaction-intensive traffic scenes on occupancy grids [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The approach represents only vehicles in the occupancy grid, their states extracted from camera inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Desquaire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [13], proposed an end-to-end object-tracking approach by incorporating directly Lidar sensor data to predict the binary grid, using recurrent neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' To incorporate ego- vehicle motion, they utilize a spatial transformer to allow internal memory of RNNs to learn environment of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Mohajerin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [14] suggested an RNN-based architecture with a difference learning method, and makes OGM pre- diction in the field of view of ego-vehicle front camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [15] proposed an encoder-decoder network architecture, along with skip connections, to make long-term DOGM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' While they collect the sensor data from an autonomous vehicle, the vehicle remains stationary and only acts as the sensor collection point at different inter- sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Itkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' proposed to use evidential occupancy grid and implement PredNet architecture for the prediction [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The approach is then carried forward to develop the double-pronged architecture [17] and attention-augmented ConvLSTM [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The latter work is able to make long-term predictions, however at turns the predictions still lose the scene structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [19] addressed the problem of OGM prediction in urban scenes by incorporating vehicles semantics in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Their proposed method de- pends on the annotated vehicle data labels available in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Contrary to the conventional Occupancy Grid Prediction, we present an allo-centric DOGM representation to predict the urban traffic scene with respect to a fixed reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Apart from the conventional recurrent representation learning approaches, we also use memory-augmented learning-based video-prediction method, in relevance to learning long-term motion context of the dynamic agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' SYSTEM OVERVIEW We discuss here the overall proposed approach for allo- centric DOGM prediction, the pipeline is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Dynamic Occupancy Grid Maps Dynamic occupancy grid maps provide a discretized rep- resentation of environment in a bird’s eye view, where every cell in the grid is independent and carries information about the associated occupancy and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' To generate DOGMs, we incorporate the Conditional Monte Carlo Dense Occupancy Tracker (CMCDOT) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' This approach associates four occupancy states to the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Each cell carries the probabilities of the cell being i) occupied and static, ii) occupied and dynamic, iii) unoccupied or free and iv) if the occupancy is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The probabilities of these four states sum to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In our work, we make use of three of these states and represent the grid as an RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The channels Red, Green and Blue represent the unknown state, dynamic state and static state respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The associated probabilities of the cell in the 3-channel DOGM grid are interpreted as the pixel values of the RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The RGB grid images can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Low probabilities in all three channels leave the grid-image black, therefore, representing free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For allo-centric grid generation, we define the grid in the world frame, close to the initial position of ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The state probabilities are initially computed in an ego- centric grid, since we use the on-board sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' To ensure that we have cell information for the complete allo-centric grid dimensions when the vehicle is dynamic and moving away from the world frame origin, a much larger ego-centric Allo-centric Video DOGM Prediction Generation NetworkDOGM is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' This information is then fused to update every cell states in the allo-centric grid in the world frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We compare the allo-centric and ego-centric grids at 4 time instants for the same scene and same grid dimensions in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In the allo-centric grid, the ego-vehicle (illustrated in the pink box) can be seen moving with respect to the grid, while it remains fixed in the ego-centric grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' It is important to note that the observable space around the ego-vehicle remains the same for both grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' However, since they are defined in different frames, the two cover different spaces in the scene at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We illustrate the common space covered by both grids since the start of the sequence, marked by yellow boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 2: Visualization of allo-centric and ego-centric grids, generated for the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The area marked by yellow lines is the common region covered by both grids up until the t-th sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The ego-vehicle is boxed in pink grid and the bus passing by is encircled in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Problem Formulation We formally define the task of predicting the scene in allo-centric DOGM representation, as sequence-to-sequence learning, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' A sequence comprises of a set of sequential grid images that capture the evolution of a given scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Let Xt ∈ R3xW xH and Yt ∈ R3xW xH be the t-th frame of the 3-channel grid-image where W and H denote the width and height respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The input sequence for the grid- image is denoted by Xt−N:t, representing N consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Given a set of input sequence, the task of the network is to predict future grid images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' output sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The target and predicted output sequences are denoted by Yt+1:t+P and ˆYt+1:t+P where P is the prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For training and testing data, the DOGMs can be generated for both the input and the target sequences, leaving behind no additional need for labelled data or human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Since the input sequences, Xt−N:t, and output sequences, Yt+1:t+P , are represented as images, this prediction task can be considered a video prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Deep Learning Prediction Architectures To study and compare the scene prediction with ego- centric and allo-centric grids, we train our datasets with different video prediction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We consider 3 networks, briefly discussed in section II-A: PredNet, PredRNN, LMC- Memory with memory alignment learning (here on referred as LMC-Memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' PredNet [8], inspired from predictive coding, makes pre- dictions based on how the predicted frames deviate from the target [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The original work tests the network on vehicle mounted camera images from Kitti dataset [21] and demon- strates the ability to capture both egocentric motion as well as motion of objects in camera images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We consider PredRNN [9] and LMC-Memory architecture [11] as the state of the art video prediction networks that aim to capture long-term dependencies and motion context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' PredRNN implements novel ST-LSTM units with a zigzag internal memory flow and proposes memory decoupling loss to discourage learning redundant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' LMC-Memory architecture, on the other hand, proposes an external memory block with its own parameters to store various motion contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The approach also offers an efficient computation method since the motion context for long-term multi-step predictions is computed only once for a given input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We study these networks capabilities to retain the occu- pancy of the static region, and the ability to predict motion of dynamic agents in DOGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Unknown Channel and Loss functions In both ego-centric and allo-centric grids, a significant part of the scene remains unobserved, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 2 (unknown channel is represented in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' This is more pronounced in the initial frames of the allo-centric grid, where the Lidar is unable to detect the farthest area from the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' While it is more relevant to learn the evolution of static and dynamic components in the scene, inclusion of unknown channel is useful for our prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' A Lidar based grid is often unable to capture the full shape of a vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For example, we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 2 how the occupied cells by the bus vary in different time steps on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' It is only in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='0s time step that a rectangular shape is observed, otherwise different parts of the bus remain unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The unknown channel at different instants also carries spatial information of the agents with respect to the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Thus, with the sequential frames and the unknown channel, we assist the network to be able to extract spatial information and learn scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The inclusion of unknown channel and emphasis on learning static and dynamic components is addressed in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Loss function L in the implemented video prediction networks is modified to carry the weighted sum of the RGB channels: L = αLR + β(LG + LB) (1) where, LR, LG and LB represent the loss for unknown (red), dynamic (green) and static channels (blue) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In order to encourage the network to learn and improve the prediction of the static and dynamic channels, α is always kept smaller than β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Dataset We study the prediction performance on the real-world NuScenes dataset [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The original dataset consists of 850 scenes for training and 150 scenes for testing, each scene is approximately 20s long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We generate the DOGM grid- based on the Lidar pointcloud and available odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For allo-centric grid, we represent the scene with respect to a fixed reference frame and a grid dimension of 60 x 60m, with a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='1m per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Each sequence starts with the ego-vehicle heading facing up, capturing the scene 10m behind and 50m ahead of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The initial pose was selected to ensure that the ego-vehicle remains within the grid for the total sequence length, even when running at a high speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For egocentric grid, we generate a grid of the same dimensions and resolution, and the ego-vehicle fixed in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Each sequence is comprised of 35 frames, a time duration of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s with DOGM grid images generated every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In total, we have 4,250 training and 750 testing sequences respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Training The input sequence Xt−9:t consists of 10 frames (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='0s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Each network is trained to make predictions ˆYt+1:t+25 for 25 future frames (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Both the allo-centric and ego- centric datasets are trained with the original parameters of the respective video prediction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For training with PredRNN and LMC Memory networks, both allo-centric and ego-centric grid images are resized to 192x192 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' PredRNN is trained with a batch size of 4 and a learning rate of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The number of channels of each hidden state is set to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The loss function is the sum of L2 and decoupling loss, and the values of α and β in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' (1) are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' LMC-Memory is trained with a learning rate of 2x10−4, memory slot is set to 100 and ConvLSTM to 4 layers for frame predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The loss function is the sum of L1 and L2 losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The values of α and β are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For training with PredNet, the grid images are resized to 160x160 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The network is set to 4 hierarchical layers with an initial learning rate of 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The loss function is the L1 loss of only the first layer, the values of α and β are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' All models are trained on Adam optimizer for 30 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' EVALUATION For evaluation, we are particularly interested in static and dynamic agents in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We discussed in section III-D, the utility of unknown regions in learning scene representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' But the unknown region occupies a big portion of the grid and, thus, in evaluation, overshadows the performance of more interesting and relevant segments: static and dynamic regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For this reason, we evaluate the dataset and network performances based on two channels of the predicted images, the blue and green channels representing static and dynamic components in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We encourage the readers to refer to the video1 for a better visualization of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='be/z-0BVM93X8c A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Quantitative Evaluation The allo-centric and ego-centric grids at any instant ob- serve different parts of the scene, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For fair comparison between them, we modify the test dataset and crop out the part of each t-frame that has not been observed until the t-th sequence by both grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Thus, for example, the part of the grids outside of the yellow dotted boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 2 are blacked out for the input sequence frames Xt−N:t as well as the target frames in the output sequence Yt+1:t+P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We measure the performances using three metrics: MSE (Mean Square Error), SSIM (Structured Similarity Indexing Method), and LPIPS (Learned Perceptual Image Patch Sim- ilarity) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' MSE is calculated by the pixel-wise difference between the ground truth and the predicted frame per channel and per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' However, with MSE, the slightest error in predicted motion can result in large errors in the ego- centric grids dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The SSIM and LPIPS metrics evaluate the prediction results based on the structural similarity and perception similarity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Lower values are better for MSE and LPIPS while higher values are better for SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Table I shows average results for the complete 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s pre- diction horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The MSE score of allo-centric grids is significantly lower compared to the one of ego-centric grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Since the complete scene transforms with respect to the ego-vehicle, the MSE is always higher in the ego-centric grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The SSIM and LPIPS scores are also significantly superior for the allo-centric grid, due to the tendency of ego- centric grids to get increasingly blurry for higher prediction horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Network MSE x 10−2(↓) SSIM(↑) LPIPS(↓) Allo-centric grid LMC-Memory 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='167 PredRNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='167 PredNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='172 Ego-centric grid LMC-Memory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='217 PredRNN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='234 PredNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='225 TABLE I: Average results with allo-centric and ego-centric grids for prediction horizon of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The allocentric grid outperforms the other in all three video prediction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 3, we plot scores of the metrics for every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s prediction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The results with allo-centric grid (shown in blue) always perform better than the ego-centric grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Among the three prediction networks, overall PredRNN performs the best with allo-centric grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' However, with the ego-centric grids (results shown in orange), PredRNN offers a good MSE score but the SSIM and LPIPS performances drop after 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' This is because PredRNN tends to make blurry and diffused predictions in the output frames;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' this helps reduce the MSE but the scene loses its structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' This is further seen in the qualitative results discussed in section V-B and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Qualitative Evaluation The prediction results between the allo-centric and ego- centric grids differ drastically when the ego-vehicle is turning Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 3: Results with the MSE(↓), SSIM(↑) and LPIPS(↓) metrics with allo-centric and ego-centric grids for input sequences of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='0s and prediction horizon up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For fair comparison, all test sequence frames were modified to only contain the scene observable in both the allo-centric and ego-centric grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The allo-centric grid (results plotted in blue) outperforms the other with all three video prediction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' (a) Allo-centric grids (b) Ego-centric grids Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 4: Qualitative results for the ego-vehicle leaving a roundabout on both allo-centric (4a) and ego-centric grids (4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The input sequence consists of 10 frames (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='0s) and output predicted sequence of up to 25 frames (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The prediction results are shown at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s instants and are magnified at the interesting spaces, marked by red box in the target(ground truth) frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The best results can be observed with LMC-Memory network with the allo-centric grid that retains the scene structure and predicts the motion of the ego-vehicle best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' at an intersection or driving on a curved road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Figure 4 shows results for a sequence where the ego-vehicle is exiting a roundabout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In this scene, while there are no other dynamic agents, the network needs to predict the behaviour of the ego- vehicle, when it is driving along the curved static segment (alluding to road structure) and is headed towards static objects/obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For the allo-centric grids, the challenge is to predict the ego-vehicle pose while the scene remains static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The best results are achieved with the LMC-Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The vehicle pose is well-predicted up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s, its orientation is adjusted so that it does not hit the static components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For the same grid, the PredRNN fails to learn and predict the behaviour resulting in false prediction of collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The ego- vehicle, while getting more blurry, diffuses into the static obstacles on the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' With the PredNet, the ego vehicle is almost already lost at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' This is expected behaviour since PredNet is ideally not aimed at long-term video predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' With all three networks, the ego- vehicle gets more blurry, however with PredRNN, the static scene also tends to get blurry at larger prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 洋LMC-ego PredRNN-ego PredNet-ego LMC-allo PredRNN-allo PredNet-alloIn the ego-centric grid, the whole scene rotates around the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' LMC-memory and PredNet significantly lose the static components ahead of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The rotation results in increasing blurriness at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' PredRNN predictions are more diffused and faint blurry cells are still visible ahead of the vehicle, even at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='5s prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In context of planning and safe navigation, this high uncertainty in the environment structure renders the prediction results unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' DISCUSSION AND FUTURE WORK In this work, we presented a novel allo-centric dynamic ocuupancy grid approach for long-term prediction of urban traffic scene, and compared it to the conventional ego- centric DOGM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We trained and tested various video prediction networks to show that allo-centric DOGM representation has superior ability to predict the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The most significant improvement is the allo-centric grid’s ability to retain the static scene structure, especially when the vehicle turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The ego-centric grid, on the other hand tends to lose the static scene, and hence the crucial information about whether the given space is occupied or free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' The results of allo-centric grids prediction with state-of- the-art PredRNN and LMC-Memory approaches have shown complementary benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' PredRNN predictions, though dif- fuse and get more blurry, are capable of maintaining agents longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' We observe that LMC-memory shows better tendency at learning behaviours in comparison to the PredRNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' It is pertinent to mention here that the two grids are still very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' In both scenarios, the observable space updates relative to the position of the vehicle in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Thus, in allo-centric grid while the grid is no more fixed to the ego-vehicle, the ego-vehicle bias remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' All three video prediction networks tested in this work address the prediction problem as deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' However, the behaviour of agents in urban traffic scene tends to be multimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' For future work, the addition of multimodal prediction capabilities in the network architecture would be interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Additionally, the incorporation of semantics in the occupancy grid such as agent type and offline road infor- mation could assist in learning behaviours and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Mozaffari, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Al-Jarrah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Dianati, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Jennings, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Mouza- kitis, “Deep learning-based vehicle behavior prediction for au- tonomous driving applications: A review,” IEEE Transactions on Intelligent Transportation Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 33–47, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Alahi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Goel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Ramanathan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Robicquet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Fei-Fei, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Savarese, “Social lstm: Human trajectory prediction in crowded spaces,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 961–971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Caesar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Bankiti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Lang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Vora, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Liong, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Xu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Kr- ishnan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Pan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Baldan, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Beijbom, “nuscenes: A multimodal dataset for autonomous driving,” arXiv preprint arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='11027, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Ettinger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Cheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Caine, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Pradhan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Chai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Sapp, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Qi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=', “Large scale interactive motion forecasting for autonomous driving: The waymo open motion dataset,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 9710–9719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' N`egre, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Rummelhard, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Laugier, “Hybrid sampling bayesian occupancy filter,” in 2014 IEEE Intelligent Vehicles Symposium Pro- ceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' IEEE, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1307–1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [6] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Rummelhard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' N`egre, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Laugier, “Conditional monte carlo dense occupancy tracker,” in 2015 IEEE 18th International Conference on Intelligent Transportation Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' IEEE, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 2485–2490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Oprea, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Martinez-Gonzalez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Garcia-Garcia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Castro- Vargas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Orts-Escolano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Garcia-Rodriguez, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Argyros, “A review on deep learning techniques for video prediction,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='org/pdf/2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='05214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='pdf [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Lotter, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Kreiman, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Cox, “Deep predictive coding networks for video prediction and unsupervised learning,” 5th International Conference on Learning Representations, ICLR 2017 - Conference Track Proceedings, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1–18, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Yu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Long, “Predrnn: A recurrent neural network for spatiotemporal predictive learning,” IEEE Transactions on Pattern Analysis and Machine Intel- ligence, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1–1, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Long, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Gao, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Yu, “Predrnn: Recurrent neural networks for predictive learning using spatiotemporal lstms,” in Advances in Neural Information Processing Systems, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Guyon, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Luxburg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Bengio, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Wallach, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Fergus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Vishwanathan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Garnett, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Choi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Kim, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Ro, “Video prediction recalling long-term motion context via memory alignment learning,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 3054–3063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Jeon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Kum, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Jeong, “Traffic Scene Prediction via Deep Learning: Introduction of Multi-Channel Occupancy Grid Map as a Scene Representation,” in 2018 IEEE Intelligent Vehicles Symposium (IV), 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 1496–1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Dequaire, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Ondr´uˇska, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Rao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Wang, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Posner, “Deep tracking in the wild: End-to-end tracking using recurrent neural networks,” The International Journal of Robotics Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 4-5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 492–512, jun 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='1177/0278364917710543 [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Mohajerin and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Rohani, “Multi-step prediction of occupancy grid maps with recurrent neural networks,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 10 600–10 608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Schreiber, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Hoermann, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Dietmayer, “Long-term occupancy grid prediction using recurrent neural networks,” in 2019 International Conference on Robotics and Automation (ICRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 9299–9305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Itkina, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Driggs-Campbell, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Kochenderfer, “Dynamic environment prediction in urban scenes using recurrent representation learning,” in 2019 IEEE Intelligent Transportation Systems Conference (ITSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 2052–2059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Available: https: //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='org/abs/1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='12374 [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Toyungyernsub, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Itkina, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Senanayake, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Kochenderfer, “Double-Prong ConvLSTM for Spatiotemporal Occupancy Prediction in Dynamic Environments,” arXiv preprint arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='09045, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='org/abs/2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='09045 [18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Lange, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Itkina, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Kochenderfer, “Attention Aug- mented ConvLSTM for Environment Prediction,” arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='09662, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [19] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Mann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Tomy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Paigwar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Renzaglia, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Laugier, “Predicting future occupancy grids in dynamic environment with spatio-temporal learning,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='org/ abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content='03212 [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Rane, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Sz¨ugyi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Saxena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Ofner, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Stober, “Prednet and predictive coding: A critical review,” in Proceedings of the 2020 international conference on multimedia retrieval, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' 233–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Geiger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Lenz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Stiller, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Urtasun, “Vision meets robotics: The kitti dataset,” International Journal of Robotics Research (IJRR), 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Isola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Efros, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Shechtman, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} +page_content=' Wang, “The unreasonable effectiveness of deep features as a perceptual metric,” in CVPR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE3T4oBgHgl3EQfUwqw/content/2301.04454v1.pdf'} diff --git a/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/2301.11424v1.pdf.txt b/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/2301.11424v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ae1030ff3cce8542daf3062058eea101d9706fd --- /dev/null +++ b/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/2301.11424v1.pdf.txt @@ -0,0 +1,2967 @@ +arXiv:2301.11424v1 [math.CT] 26 Jan 2023 +An inductive model structure for strict +∞-categories +Simon Henry and Felix Loubaton +Abstract +We construct a left semi-model category of “marked strict ∞-categories” +for which the fibrant objects are those whose marked arrows satisfy nat- +ural closure properties and are weakly invertible. The canonical model +structure on strict ∞-categories can be recovered as a left Bousfield local- +ization of this model structure. We show that an appropriate extension +of the Street nerve to the marked setting produces a Quillen adjunction +between our model category and the Verity model structure for complicial +sets, generalizing previous results by the second named author. Finally, +we use this model structure to study, in the setting of strict ∞-categories, +the idea that there are several non-equivalent notions of weak (∞, ∞)- +categories - depending on what tower of (∞, n)-categories is used. We +show that there ought to be at least three different notions of (∞, ∞)- +categories. +Contents +1 +Introduction +2 +1.1 +The street nerve as a right Quillen functor . . . . . . . . . . . . . +3 +1.2 +The two (?) notions of (∞, ∞)-categories +. . . . . . . . . . . . . +3 +2 +∞-categories and marked ∞-categories +5 +2.1 +∞-categories +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.2 +Marked ∞-categories . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.3 +Tensor product of m-marked ∞-categories . . . . . . . . . . . . . +9 +2.4 +The semi-model structure . . . . . . . . . . . . . . . . . . . . . . +12 +3 +Equations and saturations in an m-marked ∞-category. +17 +3.1 +Definitions of equations and saturations . . . . . . . . . . . . . . +18 +3.2 +Characterization of fibrant objects . . . . . . . . . . . . . . . . . +20 +3.3 +Isofibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +3.4 +Equivalences +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +3.5 +The saturated localization. +. . . . . . . . . . . . . . . . . . . . . +27 +4 +Comparison with other model structures +29 +4.1 +Truncation functors +. . . . . . . . . . . . . . . . . . . . . . . . . +29 +4.2 +Comparison with the folk model structure on ∞-Cat . . . . . . . +36 +4.3 +The folk model structure vs the limit of the π-tower +. . . . . . . +38 +4.4 +Complicial sets and stratified Street nerve . . . . . . . . . . . . . +42 +1 + +1 +Introduction +In the present paper, we introduce (in Section 2.2) a category ∞-Catm of “m- +marked (strict) ∞-categories” for m ∈ N ∪ {∞}. The objects of ∞-Catm are +strict ∞-categories, with, similarly to stratified simplicial sets, some arrows be- +ing “marked”. The marked arrows are required to be closed under composition, +and all identities arrows as well as all arrows of dimension > m are marked. This +category ∞-Catm is equipped with two monoidal closed structures denoted → +and +∼ that are both the Gray-Crans tensor product on the underlying strict +∞-categories but act differently on markings. These two monoidal structures +are meant to respectively be models for the“lax-Gray tensor product” and the +“pseudo-Gray tensor product”. +Our main result is the construction of a model structure1 on ∞-Catm similar +to the canonical (or “Folk”) model structure on strict ∞-category from [19]: +1.1 Theorem. There is a combinatorial left semi-model structure on the cate- +gory ∞-Catm of m-marked ∞-categories such that: +• This model structure is monoidal for both tensor products +∼ and → (from +Section 2.3). +• The cofibrations are the map that are cofibrations of the canonical model +structure between the underlying ∞-categories. +• The fibrant objects are the marked ∞-categories in which all marked arrows +admit marked weak inverses, and in which if there is a marked arrow a → b +then a is marked if and only if b is marked. +• Fibrations between fibrant objects are the “isofibrations” (as defined in +Section 3.3). +• Weak equivalences between fibrant objects are “equivalence of marked ∞- +categories”(as defined in Section 3.4). +This model structure is a model for strict “(∞, m)-categories” where “invert- +ibility” or arrows of dimension > m is taken in a weak sense. The existence of +this model structure is established in Section 2.4, but some of its properties, in +particular, the characterization of fibrant objects and fibrations between fibrant +objects will only be established in Section 3. +We also consider two left Bousfield localizations of this model structure: +• The saturated inductive model structure, studied in Section 3.5, whose +fibrant objects are the ∞-categories in which every arrow which is weakly +invertible up to marked arrows is also marked. +• The coinductive model structure, studied in Section 4.2, whose fibrant +objects are the ∞-categories in which every coinductively invertible (see +Definition 4.16) arrow is marked. +This second localization is equivalent2 to the canonical model structure on +∞-categories from [19]. +The motivations to introduce this model structure come from two different +lines of investigations that we will explain separately below: +1We use the term “model category” as a generic name for all sorts of model categories +(Quillen model categories, semi-model categories, weak model categories, etc...) +2Though not through a Quillen equivalence. +2 + +1.1 +The street nerve as a right Quillen functor +In [20], the second named author has shown that the Street nerve of a strict +∞-category can be made into a complicial set by defining the “thin” simplexes +as being those whose top dimensional arrows are weakly invertible. +From there, it is natural to ask whether this stratified version of the Street +nerve, also preserves fibrations, and hence is a morphism of categories of fibrant +objects (and this will be shown in the present paper as Proposition 4.51). +In fact, more generally, one could ask if it is possible to make this version +of the Street nerve into a right Quillen functor (for the Verity model structure +on complicial sets from [30]). This is not directly possible simply because this +stratified Street nerve is not a right adjoint functor. The solution to this problem +is to work with marking on both sides: The usual Street nerve from strict ∞- +categories to simplicial sets is a right adjoint functor, and one can extend it to a +right adjoint functor from marked ∞-categories to “marked” simplicial sets (or +rather stratified simplicial sets to follow the terminology of [30]). In Section 4.4 +we show that this functor is indeed a right Quillen functor. +This right Quillen functor from marked ∞-categories to stratified simplicial +sets is meant to be a model for the forgetful functor from strict (∞, ∞)-categories +to weak (∞, ∞)-categories. In particular, the corresponding left Quillen functor +from stratified simplicial sets to marked ∞-categories is a model for the more +mysterious “strictification functor”, sending weak ∞-categories to strict ∞- +categories. +At the level of ∞-groupoids, this strictification functor corresponds essen- +tially to (non-abelian) homology, through the equivalence between strict ∞- +groupoids and crossed chain complexes ([7]) which is well-known to be a conser- +vative functor by Whitehead’s theorem for homology. The first named author +has conjectured [17] that more generally this strictification functor should be +conservative on weak (∞, m)-categories for all m. +This allows us to state a +concrete version of this conjecture here: +1.2 Conjecture. The left Quillen functor | |: sSetm → ∞-Catm from Sec- +tion 4.4 reflects weak equivalence between cofibrant objects. +1.2 +The two (?) notions of (∞, ∞)-categories +C.Schommer-Pries and C.Rezk have independently argued ([16]) that there +should be more than one notion of weak (∞, ∞)-categories. More precisely, +they both arrive at the conclusion that even if one accepts (which seems to +be a clear consensus nowadays) that there is only one notion of weak (∞, n)- +categories for finite n, there are at least two different ways to build a notion of +(∞, ∞)-categories out of it. +Before we go into further details, we should say that the following discussion +is mostly informal and speculative and most of it has not been formalized in +any models - in fact, one motivation for the present paper is to formalize some +of it in the context of strict ∞-categories. +First, let us go over the argument put forward by Rezk and Schommer- +Pries, or at least how we understand it: The forgetful (or inclusion) functor +from (∞, n)-categories to (∞, n + 1)-categories is supposed to have both a left +adjoint πn, which freely adds inverses to all (n + 1)-arrows and a right adjoint +τn which remove all non-invertible (n + 1)-arrows. +3 + +This allows to produce two different towers: +(∞, 0)-Cat +π0 +← (∞, 1)-Cat +π1 +← (∞, 2)-Cat +π2 +← . . . +πn−1 +← (∞, n)-Cat +πn +← . . . +(∞, 0)-Cat +τ0 +← (∞, 1)-Cat +τ1 +← (∞, 2)-Cat +τ2 +← . . . +τn−1 +← (∞, n)-Cat +τn +← . . . +and one can take the projective limit of either of these two towers to give a +definition of what is an (∞, ∞)-category. +If one takes the limits of the π-tower then one can see that an arrow that +is “coinductively” invertible (see Definition 4.16) has to be considered invert- +ible. To be precise, we mean that if F: X → Y is a morphism in the limit of +the π-tower which admits an inverse up to a coinductively invertible natural +transformation then F is an equivalence. +The situation in the limit of the τ-tower however is fairly different: Given an +(∞, ∞)-category in this sense, it corresponds to a collection of (∞, n)-categories +Xn such that Xn ≃ τnXn+1, and an n-arrow corresponds to an n-arrow of Xn +(or of Xk for k > n). In this setting one has an intrinsic notion of equivalence: +an n-arrow is said to be an equivalence if it belongs to Xn−1 (equivalently if it +is invertible in the (∞, n)-category Xn). In this setting, coinductively invertible +arrows do not have to be invertible if none of the higher cells witnessing the +coinductive invertibility are not themselves invertible. +To clearly show that the two are different, one can for example consider the +(∞, ∞)-category of cobordisms. In the limit of the τ-tower one can define it by +taking Xn to be the (∞, n)-categories of cobordisms. In this (∞, ∞)-category, +every arrow has a dual, so it follows from a result of E.Cheng (see [9]) that every +arrow in the cobordisms (∞, ∞)-category is coinductively invertible, although +there are many non-invertible n-arrows in Xn for all n. +Hence, if one were +trying to define Xn in the limit of the π-tower, it would be equivalent to an +∞-groupoid. +Using our model structure of marked strict ∞-category, we will make these +two constructions formal in the context of strict ∞-category. This is of course +only meant to be a toy model for the case of weak ∞-categories, but it is already +interesting, and it will show that the picture above while correct, needs to be +refined a little. +First, we will show in Section 4.1 that our model structure on ∞-Catm +for m = ∞ corresponds to the limit of τ-tower as above. More precisely, we +will show that it is Quillen equivalent to an appropriate homotopy limit of the +∞-Catm for m < ∞ using the τn functor as transition functors. +The notion of homotopy limit of a tower of model structure we are using has +been introduced in [6], and we will use their construction of the homotopy limit. +Here there is a small gap we should disclaim: [6] only develops the theory of such +limits for Quillen model categories and not semi-model categories, and we will +apply their construction to our left semi-model categories directly. In order for +our argument to be complete despite this, we will prove that the construction +from [6] does yield to a left semi-model category, but we will not reprove that +it corresponds to a homotopy limit as in [6, Theorem 5.1]. However, it should +be noted that in practice, the argument of [6] seems to carry over to our setting +with almost no changes, so this gap is not really a concern. +4 + +In Section 4.2 we will show that the folk model structure is equivalent to the +left Bousfield localization of our model structure which corresponds to turning +all coinductively invertible arrows into equivalences. +However, we will also show in Section 4.3, that the folk model structure is +not equivalent to the limit of the π-tower. It is unclear if the limit of the tower +of πn corresponds to further localization of our model structure, or is something +entirely different, but we find that the argument we will give in Section 4.3 to +distinguish between the folk model structure and the limit of π-tower shows +that this limit is exhibiting behaviors that are not really expected from a notion +of (∞, ∞)-categories, or at least are not typical of any known model of ∞- +categories. +Coming back to the world of weak (∞, ∞)-categories, this suggests that +the two most interesting notions of weak (∞, ∞)-categories are the limit of +τn tower, which corresponds to an “inductive” notion of equivalences, and its +localization that turn the coinductive equivalence into equivalences, but this +localization should be different from the limit of the πn-tower which might not +be an interesting notion of (∞, ∞)-categories. What we mean here is that we are +not aware of any attempt of giving a concrete definition of (∞, ∞)-categories +that seems to produce something that could be equivalent to this limit. All +definitions we have seen can be reasonably conjectured to be equivalent to either +the limit of the τn tower or to its “coinductive” localization. +2 +∞-categories and marked ∞-categories +2.1 +∞-categories +A globular set is a presheaf on the globular category G: +D0 +D1 +D2 +D3 +D4 . . . +i+ +0 +i− +0 +i+ +1 +i− +1 +i+ +2 +i− +2 +i+ +3 +i− +3 +With the relations iǫ +ni+ +n−1 = iǫ +ni− +n−1 for all n > 0 and ǫ ∈ {+, −}. We also +denote by iǫ +k the map Dk → Dn for k < n obtained by composing any string of +arrow ending with iǫ +k. These and the identity arrows are the only maps in the +category G. +If X is a globular set, one denotes by Xn the set X(Dn) whose elements are +called n-arrows. The map Xn → Xk induced by iǫ +k: Dk → Dn is denoted by πǫ +k. +2.1 Definition. An ∞-category is a globular set X together with operations +of compositions +Xn ×Xk Xn → Xn +(0 ≤ k < n) +which associates to two n-arrows (x, y) verifying π+ +k (x) = π− +k (y), one n-arrow +x#ky, as well as identities +Xn → Xn+1 +associating to an n-arrow x, an (n + 1)-arrow Ix, and satisfying the following +axioms: +5 + +(1) ∀x ∈ Xn, πǫ +n(Ix) = x. +(2) π− +k (x#ny) = π− +k (x) and π+ +k (x#ny) = π+ +k (y) whenever the composition is +defined and k ⩽ n. +(3) πǫ +k(x#ny) = πǫ +k(x)#nπǫ +k(y) whenever the composition is defined and k > +n. +(4) x#nIπ+ +n x = x and Iπ− +n x#nx = x. +(5) (x#ny)#nz = x#n(y#nz) as soon as one of these is defined. +(6) If k < n +(x#ny)#k(z#nw) = (x#kz)#n(y#kw) +when the left-hand side is defined. +A morphism of ∞-categories is a map of globular sets commuting with both +operations. The category of ∞-categories is denoted ∞-Cat. +2.2 Definition. An (n + 1)-arrow c in an ∞-category is said to be trivial, or +an identity arrow, if there exists an n-cell d such that c = Id. +2.3 Example. By abuse of notation, we also denote Dn the ∞-category that +admits for any k < n only two k-non-trivial arrows, denoted e− +k and e+ +k , and a +single non-trivial n-arrow, denoted en verifying : +π− +l (eǫ +k) = e− +l +π+ +l (eǫ +k) = e+ +l +for l ≤ k < n +π− +l (en) = e− +l +π+ +l (en) = e+ +l +for l ≤ n +The ∞-category ∂Dn is obtained from Dn by removing the n-arrow en. We +thus have a morphism +in: ∂Dn → Dn. +Note that ∂D0 = ∅ +2.4 Definition. If X is an ∞-category, we define the globular set ΣX, called +the suspension of X, by the formula +(ΣX)0 = {a, b} +(ΣX)n+1: = Xn ∪ {Ina, Inb} +where In +a (resp. In +b ) is the n-times iterated unity of a (resp. of b). Moreover, +ΣX inherits from X a structure of ∞-category. +Eventually, for an integer n, we define the ∞-category ΣnX, called the n- +suspension of X, as the n-times iterated suspension of X. +Next, we define the notion of polygraphs, first introduced under the name +“computads” by R. Street in [28] for 2-categories, with the general notion being +hinted at in [29]. As far as we know the first formal introduction of polygraphs +in the literature is in [25] and independently in [8], where the name “polygraphs” +was introduced. Here we will exploit that the category of polygraphs identifies +with a (non-full) subcategory of ∞-Cat to give a shorter definition. We refer +to the references above for a more complete introduction. +6 + +2.5 Definition. +• We say that an ∞-category X is a polygraph if it can be constructed +from the empty ∞-category by freely adding arrows with specified source +and target. That is if X can be obtained as a transfinite composition +∅ = X0 → X1 → · · · → Xi → Colim Xi = X where for each i, the map +Xi → Xi+1 is a pushout of Y × ∂Dn → Y × Dn+1. +• An arrow of a polygraph is said to be a generator if it is one of the arrows +that has been freely added at some stage. +• A morphism of ∞-categories between two polygraphs is said to be a mor- +phism of polygraphs or a polygraphic morphism if it sends each generator +to a generator. +• An n-polygraph is a polygraph whose generators are all of dimension ⩽ n. +2.6 Remark. Generators of a polygraph can be shown to be exactly the arrows +that cannot be written as a composite in a non-trivial way, so the notion of +generator does not depend on the choice of the presentation of X, and any +isomorphism between polygraphs is automatically polygraphic. +2.7 Example. The only n-polygraphs for n < 0 is the empty ∞-category, the +category of 0-polygraphs is equivalent to the category of sets and corresponds +to discrete ∞-categories, the category of 1-polygraphs (and polygraphic mor- +phisms between them) is equivalent to the category of directed graphs, and they +corresponds to categories that are free on a graph. +We will sometimes distinguish between a polygraph seen as an object of +the category of polygraphs and polygraphic morphisms, and the corresponding +∞-category, which we call the free ∞-category on the polygraph. +2.8 Remark. Each arrow in a polygraph can be written as an iterated compos- +ite of the generators (not necessarily in a unique way). For an n-arrow f, the set +of generators of dimension n that appear in such an expression (and even the +number of times they appear) is the same for all such expressions. We will say +that an n-generator appears in an n-arrow if it appears in any such expression. +2.9 Construction. The category ∞-Cat admits a closed monoidal structure, +called the Gray tensor product or Crans-Gray tensor product, which we denote +as +∞-Cat × ∞-Cat +→ +∞-Cat +X, Y +�→ +X ⊗ Y +Its explicit construction is very involved and we will assume the reader is already +familiar with it. It was first introduced by S. Crans in his Ph.D. thesis [10]. +We refer to [1] for an introduction to this tensor product close to its original +definition, and to [27] for a more modern account. The proof of the existence of +this monoidal structure in [27] contains some gaps that have been fixed in [4]. +It is easy to see from either of these definitions that Dn ⊗ Dm has a unique +non-trivial arrow of dimension n + m. If f and g are respectively an n-arrow +of X and an m-arrow of Y , which corresponds to morphisms f: Dn → X and +g: Dm → Y , we denote by f ⊗ g the m + n arrow of X ⊗ Y obtained as the +image of this non-trivial (n+m)-arrow by the functor f ⊗g: Dn ⊗Dm → X ⊗Y . +We recall from [3]: +7 + +2.10 Proposition. If X and Y are polygraphs then X ⊗ Y is also a polygraph. +The generators of X ⊗ Y are the arrow of the form x ⊗ y where x and y are +respectively generators of X and Y . +Finally, we recall from [19] that ∞-Cat carries a model structure, called the +folk model structure in which every object is fibrant and where the generating +cofibrations are the ∂Dn → Dn. Its weak equivalences are a natural class of +equivalence of ∞-categories that generalizes the equivalences of ordinary cate- +gories. It was shown in [23] that the cofibrant objects are exactly the polygraphs +and it also follows from this that the cofibrations between cofibrant objects are +the polygraphic inclusions. It was shown in [3] that this model structure is a +monoidal model structure for the Gray tensor product. +2.2 +Marked ∞-categories +For the rest of the article, we fix an m ∈ N ∪ {∞} +2.11 Definition. An m-marked ∞-category is an ∞-category X, together with +a set M ⊂ � +k>0 X(k) of arrows of positive dimension called marked arrows such +that: +• All identity arrows Ix are marked. +• All arrows of dimension strictly superior to m are marked. +• If x and y are marked n-arrows and x#ky is defined, then x#ky is marked. +A morphism of m-marked ∞-categories is a morphism between the under- +lying ∞-categories that sends marked arrows to marked arrows. The category +of m-marked ∞-categories is denoted ∞-Catm. +Note that if m = ∞, then the second condition of the definition simply +disappears, this is the main case we are interested in. +2.12 Example. If X is an ∞-category we denote by X# the m-marked ∞- +category (X, X>0) where all arrows of positive dimension are marked. We denote +by X♭ the m-marked ∞-category where only identity arrows and k-arrows for +k > m are marked. +2.13 Construction. If X is an ∞-category and M ⊂ � +k>0 Xk is a set of +arrows of X, we denote by M the smallest set of arrows such that M ⊂ M +and (X, M) is an m-marked ∞-category. That is M is the reunion of the set of +arrows of dimension strictly superior to m and the set of all n-arrows that can +be written as iterated composites of n-arrows in M and arrows of the form Ix +for x an (n − 1)-arrow. For example X♭ = (X, ∅). +2.14 Construction. The category of m-marked ∞-categories has all colimits, +and they are easily described in terms of colimits of ∞-category and of Con- +struction 2.13: if (Xi, Mi)i∈I is a diagram of m-marked ∞-category indexed by +a category I then: +Colim +i∈I (Xi, Mi) = +� +Colim +i∈I +Xi, ∪ifi(Mi) +� +where fi denotes the canonical map fi: Xi → Colimi∈I Xi and fi(Mi) is simply +the set of arrows of the form fi(x) for x ∈ Mi. +8 + +This is easily shown by checking that the right-hand side has the universal +property of the colimit. +2.15 Definition. A special m-marked polygraph is an m-marked ∞-category of +the form (X, M) where X is free on a polygraph and M only contains generators +of X. +2.16 Proposition. If (X, M) is a special m-marked polygraph, then an n-arrow +f is in M if and only if n > m or if all the generating n-arrows that appear in +f are in M. +Proof. An arrow satisfying this condition is a composite of marked n-arrows and +identities of lower dimensional arrows, so it has to be in M. Conversely, this +set of arrows contains M and all identities (as no n-dimensional arrows appear +in their expression) and is closed under composition. +2.3 +Tensor product of m-marked ∞-categories +In this section we construct two monoidal closed structures on the category of +m-marked ∞-categories, respectively called the pseudo-Gray tensor product +∼ +and the lax-Gray tensor product +→. +Both are obtained by putting different +markings on the Gray tensor product from Construction 2.9. For example, the +lax-Gray tensor product D1 → D1 is C♭ +1 where C1 is the polygraph +C1 = + + + +• +• +• +• + + + +while D1 ∼ D1 is the special m-marked polygraph (C1, D) where D only contains +the unique 2 dimension generator of C1. So, unless m = 0 or m = 1, the two +tensor products are distinct. At the derived or homotopy theoretic level, the +pseudo-Gray tensor product should correspond to the cartesian product. +The formal definition goes as follows +2.17 Construction. Given two m-marked ∞-categories (X, M) and (Y, N) we +define two sets of arrows in X ⊗ Y : +• M → N is the set of arrows of the form x ⊗ y ∈ X ⊗ Y where either x ∈ M +or y ∈ N. +• M ∼ N contains all arrows in M → N together with all arrows of the form +x ⊗ y with x and y both of dimension > 0. +Note that M → N and M ∼ N are not marking on X ⊗Y : they are not stable +under composition. So we define: +(X, M) → (Y, N) = (X ⊗ Y, M → N) +(X, M) ∼ (Y, N) = (X ⊗ Y, M +∼ N) +We will show in Lemma 2.37 that both make the category of m-marked +∞-categories into a monoidal closed category. +In order to show this, it is convenient to introduce the following notations: +9 + +2.18 Notation. For A and B subsets of arrows in ∞-categories, we denote by +A ⊗ B the set of arrows of the form a ⊗ b ∈ X ⊗ Y for a ∈ A and b ∈ B. For +X and ∞-category, we denote by X⩾0 the set of all arrows of X and by X>0 +the set of all arrows of dimension > 0. We can hence, for (X, M) and (Y, N) to +m-marked ∞-category rewrite the definitions above as: +M → N += +(M ⊗ Y⩾0) ∪ (X⩾0 ⊗ N) +M +∼ N += +(M → N) ∪ (X>0 ⊗ Y>0) += +(M ⊗ Y⩾0) ∪ (X⩾0 ⊗ N) ∪ (X>0 ⊗ Y>0) +By definition of the Gray tensor product, we have the following result: +2.19 Lemma. Let X and Y be two ∞-categories, then +X⩾0 ⊗ Y⩾0 = (X ⊗ Y )⩾0 +X>0 ⊗ Y⩾0 ∪ X⩾0 ⊗ Y>0 = (X ⊗ Y )>0 . +That is X ⊗ Y is generated under composition by arrows of the form x ⊗ y, +and the arrows of dimension > 0 of X ⊗ Y are generated under compositions +by arrows of the form x ⊗ y with x or y of dimension > 0 +2.20 Lemma. Let X be an ∞-category and M, N two subsets of arrows of X +then: +M ∪ N = M ∪ N = M ∪ N = M ∪ N +Proof. This is straightforward. +2.21 Lemma. Let X, Y be two ∞-categories and M ⊂ X⩾0 and N ⊂ Y⩾0. +Then: +M ⊗ N = M ⊗ N = M ⊗ N = M ⊗ N +Proof. We will only show the equality M ⊗ N = M ⊗ N. The equality M ⊗ N = +M ⊗ N is proved in the exact same way and the last equality follows immedi- +ately by applying the result to M and N. We will also only proves the results +for m = ∞, the case of a general m follows immediately as it marks all arrow of +dimension > m on each side of these equalities. The evident inclusion M ⊂ M +implies M ⊗ N ⊂ M ⊗ N, so it is then enough to show that M ⊗ N ⊂ M ⊗ N. +Let K be the set of arrows k in X such that k ⊗ n ∈ M ⊗ N for all n ∈ N. We +need to show that K is closed by identity and composition to finish the proof. +If k = Ix, then k ⊗ n = Ix⊗n ∈ M ⊗ N. Let now k, k′ ∈ K of dimension n such +that k#ik′ is defined. They are encoded by a map Dn +� +Di Dn → X and let +y ∈ N be an arrow of dimension m of Y , encoded by a map Dm → Y . +Together these induced a map e: +� +Dn +� +Di Dn +� +⊗Dm → X⊗Y . +� +Dn +� +Di Dn +� +⊗ +Dm is a polygraph of dimension m + n with only two generating arrows of +maximal dimensions that are sent to k ⊗ y and k′ ⊗ y, which are by hypothesis +in M ⊗ N. +Now the arrow corresponding to (k#ik′) ⊗ y in +� +Dn +� +Di Dn +� +⊗ Dm is in +M ⊗ N as all the top dimensional generators that appear in it are in M ⊗ N. +We have proved that k#ik′ ⊗ y ∈ M ⊗ N for all y ∈ N, hence k#ik′ ∈ K and +this concludes the proof. +10 + +2.22 Lemma. Let X, Y be two ∞-categories, M ⊂ X⩾0 and N ⊂ Y⩾0. Then +we have +M → N += +M → N +M +∼ N += +M +∼ N. +Proof. Given the formula for M → N and M +∼ N from Notation 2.18, this is a +direct consequence of Lemma 2.20 and Lemma 2.21. +2.23 Lemma. Let X, Y, Z be three ∞-categories, M ⊂ X>0, N ⊂ Y>0 and +P ⊂ Z>0. Then we have +(M → N) → P += +M → (N → P) +(M +∼ N) ∼ P += +M +∼ (N +∼ P) +Proof. We begin with the first equality. Let +E: = (M ⊗ Y⩾0 ⊗ Z⩾0) ∪ (X⩾0 ⊗ N ⊗ Z⩾0) ∪ (X⩾0 ⊗ Y⩾0 ⊗ P) . +The lemmas 2.19, 2.20 and 2.21 implies the following equalities: +E += +M ⊗ Y⩾0 ⊗ Z⩾0 ∪ X⩾0 ⊗ (N ⊗ Z⩾0 ∪ Y⩾0 ⊗ P) += +M ⊗ (Y ⊗ Z)⩾0 ∪ X⩾0 ⊗ (N → P) += +M → (N → P) +A very similar computation also shows that E = (M → N) → P, which concludes +the proof of the first equality. +For the second equality, we define +F: = (X⩾0 ⊗ Y>0 ⊗ Z>0) ∪ (X>0 ⊗ Y⩾0 ⊗ Z>0) ∪ (X>0 ⊗ Y>0 ⊗ Z⩾0) +The second equality of Lemma 2.19 implies that: +F = Xk⩾0 ⊗ Y>0 ⊗ Z>0 ∪ X>0 ⊗ (Y ⊗ Z)>0 +and then that +E ∪ F += +M ⊗ (Y ⊗ Z)⩾0 ∪ X⩾0 ⊗ (N +∼ P) ∪ X>0 ⊗ (Y ⊗ Z)>0 += +M +∼ (N +∼ P) +and here again, a similar computation shows E ∪ F = (M +∼ N) ∼ P, which +concludes the proof. +2.24 Lemma. Let X be an ∞-category, M ⊂ X>0. +Then the empty set, +considered as a subset of the ∞-category D0, verifies (up to the identifications +D0 ⊗ X ≃ X ⊗ D0 ≃ X): +∅ → M = M → ∅ = M +∅ ∼ M = M +∼ ∅ = M +Proof. The first equality is a straightforward application of the definition of →. +For the second case, we also use that all arrows of (D0)>0 ⊗ X>0 are identities +and so all belong to M. +11 + +2.25 Proposition. Both the lax-Gray tensor product +→ and the pseudo-Gray +tensor product +∼ as defined above are monoidal structures on the category of +m-marked ∞-categories. In both cases the forgetful functor to ∞-categories is +monoidal and their unit is D♭ +0 = D# +0 . +Proof. Note that D♭ +0 = D# +0 = (D0, ∅) as all arrows of D0 of dimension > 0 are +identities. +The proposition exactly says that the structural map (associativity and unit +isomorphism) of the Gray tensor product of ∞-categories preserves the marking +we specified on the tensor product. +For the unit, let (X, M) be an m-marked ∞-category. The Lemmas 2.21 +and 2.24 imply that +(X, M) → (D0, ∅) += +(X ⊗ D0, M → ∅) += +(X, M) +(X, M) ∼ (D0, ∅) += +(X ⊗ D0, M +∼ ∅) += +(X, M) +and +(D0, ∅) → (X, M) += +(D0 ⊗ X, ∅ → M) += +(X, M) +(D0, ∅) ∼ (X, M) += +(D0 ⊗ X, ∅ ∼ M) += +(X, M). +For the associativity isomorphism, let (X, M), (Y, N) and (Z, P) be three ∞- +categories. Lemma 2.21 implies that +� +(X, M) → (Y, N) +� +→ (Z, P) += +(X ⊗ Y ⊗ Z, (M → N) → P) +� +(X, M) ∼ (Y, N) +� +→ (Z, P) += +(X ⊗ Y ⊗ Z, (M +∼ N) → P) +and +(X, M) → � +(Y, N) → (Z, P) +� += +(X ⊗ Y ⊗ Z, M → (N → P)) +(X, M) ∼ � +(Y, N) → (Z, P) +� += +(X ⊗ Y ⊗ Z, M +∼ (N → P)). +Lemma 2.23 shows that these two marking on X ⊗ Y ⊗ Z, in the lax and +the pseudo case, coincide. +2.26 Proposition. The pseudo and lax-Gray tensor product → and ∼ preserves +colimits in each variable. +In particular, as ∞-Catm is locally presentable, this immediately implies +that both tensor products are closed monoidal structures. +Proof. It follows from the fact that the Gray tensor product ⊗ preserves colimits +in each variables, the description of colimits of m-marked ∞-category given in +Construction 2.14 and Lemma 2.21. +2.4 +The semi-model structure +In this section, we will construct a left semi-model structure on the category +∞-Catm. +2.27 Definition. We define the set I = Im ∪ Ia to be our set of generating +cofibrations in ∞-Catm where: +Ia = {in: ∂Dn → Dn, |n ⩾ 0} +12 + +Im = {Dn → (Dn, {en}) +, n ⩾ 0} +An arrow in ∞-Catm is said to be a trivial fibration if it has the right lifting +property against all arrows in I. An arrow in ∞-Catm is said to be a cofibration +if it has the left lifting property against all trivial fibration. +2.28 Remark. It immediately follows from the small object argument that +every arrow can be factored into a cofibration followed by a trivial fibration +and that all cofibrations are retracts of transfinite compositions of pushouts of +arrows in I. +2.29 Remark. An arrow π: X → Y has the right lifting property against all +arrows in Ia if its image by the forgetful functor to ∞-Cat is a trivial fibration, +that is if for every pair of parallel n-arrows u, v in X, the map HomX(u, v) → +HomY (π(u), π(v)) is surjective. +π has the right lifting property against all arrows in Im if and only for every +arrow f ∈ X such that π(f) is marked in Y , f is marked in X. A trivial fibration +is a map that has both these properties. +2.30 Remark. The cofibrant objects of ∞-Catm are exactly the m-marked ∞- +categories whose underlying ∞-category is free on a polygraph, with any possible +marking on them (not just the special markings of 2.15). Indeed, transfinite +compositions of pushouts by arrows in Ia only starting from the empty ∞- +category exactly give all polygraphs with no markings on them. Pushouts by +Im are simply changing the marking and can make any arrow marked, so by also +taking pushouts by arrows in Im one obtains all polygraphs with any possible +marking on them. Finally, it was shown in [23] that polygraphs are closed under +retract in ∞-Cat, so they constitute all cofibrant objects. +The pushout-product, or corner-product (sometimes also called Leibniz prod- +uct) f ˆ +→ g and f ˆ +∼ g is defined as usual: if f: X → Y and g: A → B are two +arrows in ∞-Catm, then f ˆ +→ g is the canonical arrow: +X → B +� +X →A +Y +→ A → Y +→ B +and f ˆ +∼ g is the canonical arrow +X ∼ B +� +X ∼ A +Y +∼ A → Y +∼ B +We refer to the appendix of [18] for the general theory of pushout products and +their formal properties. +2.31 Proposition. If f and g are two cofibrations in ∞-Catm then f ˆ +→ g and +f ˆ +∼ g are both cofibrations. +Proof. By the usual properties of the corner-product, it is enough to check this +when f and g are generating cofibrations. If f and g are both in Ia, then f → g +has no marked arrows in either its domain or codomain and coincides with the +corner-product f ˆ⊗ g in ∞-Cat, which has been shown to be a cofibration in +[3]. f +∼ g is the same except that some arrows are marked, but we can always +add these marking by taking additional pushouts by arrows in Im, so it is again +a cofibration. +13 + +The forgetful functor ∞-Catm → ∞-Cat is monoidal for both tensor prod- +uct and preserves colimits, so it preserves the corner-product. In particular, if +either f or g is in Im then it is sent to isomorphisms by this forgetful functor +and hence f ˆ +→ g and f ˆ +∼ g induces isomorphisms between their underlying ∞- +categories. Now, if f: (X, N) → (X, M) is a morphism in ∞-Catm that induces +an isomorphism on underlying ∞-categories, then it is a pushout of arrows in +Im: one simply needs to take such pushout to make all arrows in M marked. +2.32 Construction. We define I: = D♯ +1 = (D1, {e1}). It is the ∞-category with +two objects, e− +0 and e+ +0 and a marked arrow e1: e− +0 → e+ +1 . We denote by j− and +j+ the two maps D0 → I corresponding respectively to the two objects e− +0 and +e+ +0 . This gives a diagram: +D0 +� +D0 ֌ I → D0 +Which will play the role of the interval object for our semi-model structure on +∞-Catm. +We will take as a set of “generating anodyne cofibrations” (also called a +“pseudo-generating set of trivial cofibrations”) the set of maps of the form j+ ˆ +∼ i +where i is a generating cofibration, more precisely: +2.33 Definition. +• We say that an arrow in ∞-Catm is a naive fibration if it has the right +lifting property against all arrows of the form j+ ˆ +∼ i, where j+: D0 → I is +as in Construction 2.32, and i is one of the generating cofibrations as in +Definition 2.27. +• We say that an arrow in ∞-Catm is an anodyne cofibrations if it has the +right lifting property against all naive fibrations. +• We say that a cofibration in ∞-Catm is acyclic if it has the lifting property +against all naive fibrations between (naively) fibrant objects. +• We say that a map in ∞-Catm is a fibration if it has the right lifting +property against all acyclic cofibrations. +As before, it immediately follows from the small object argument that every +arrow factors as an anodyne cofibration followed by a naive fibration, and all +anodyne cofibration are retracts of transfinite compositions of pushouts of the +“generating anodyne cofibrations”. +2.34 Remark. It immediately follows from Proposition 2.31 that, as j+ is a +cofibration, all maps of the form j+ ˆ +∼ i are cofibrations. In particular, all trivial +fibrations are also naive fibrations and all anodyne cofibrations are cofibrations. +2.35 Proposition. Acyclic cofibrations and fibrations form a cofibrantly gen- +erated weak factorization system on ∞-Catm. An object is “naively fibrant” if +and only if it is fibrant and more generally an arrow between fibrant objects is +a fibration if and only if it is a naive fibration. +Proof. This is a direct application of the results of Section 4 of [15]. Starting +from the premodel structure on ∞-Catm whose weak factorization systems are +(cofibrations, trivial fibrations) and (anodyne cofibrations, naive fibrations), we +obtain the one with (cofibrations, trivial fibrations) and (acyclic cofibrations, +fibrations) as its “left saturation” L(∞-Catm) in the sense of Theorem 4.1 of +[15]. All the claim in the proposition follows from this Theorem 4.1. +14 + +2.36 Remark. Note that replacing ˆ +∼ by ˆ +→ in 2.33 would not change the +definition. +Indeed, if X = Y ♯ is an m-marked ∞-category whose arrows of +dimension > 0 are all marked then for any m-marked ∞-category Z one has +X +∼ Z = X → Z. As this applies to both the domain and the co-domain of j+ +it follows that j+ ˆ +∼ i = j+ ˆ +→ i. +Also, the reader should not be worried about the use of j+ in Definition 2.33 +rather than j− or both j− and j+. While putting j− or both j− and j+ instead +of j+ would change the definition of naive fibrations and anodyne cofibrations, +this does not affect the definition of (naive) fibrations between fibrant objects, +hence the acyclic cofibrations and fibrations would not be changed. Indeed, once +the existence of a (monoidal) model structure is established, it follows that j− +is acyclic by 2-out-of-3, and hence all the maps j− ˆ +∼ i = j− ˆ +→ i are also acyclic +cofibrations. +2.37 Lemma. If f is an anodyne (resp. acyclic) cofibration and g is a cofibra- +tion then f ˆ +∼ g and f ˆ +→ g are anodyne (resp. acyclic). +Proof. To get the result for “anodyne cofibrations” it is enough to prove it for +the generating anodyne cofibrations. Let i be one of the generating cofibrations +and f = j+ ˆ +∼ i′ be one of the generating anodyne cofibrations. We have f ˆ +∼ i = +j+ ˆ +∼ (i ˆ +∼ i′). +As i′ ˆ +∼ i is a pushout of generating cofibrations i1, . . . , ik by +Proposition 2.31 it follows that j+ ˆ +∼ (i ˆ +∼ i′) is a pushout of the j+ ˆ +∼ ik and +hence is an anodyne cofibration. +The result for acyclic cofibrations follows from formal properties of the +pushout product: it follows that if i is a cofibration and p is a naive fibra- +tion then the (right) pullback exponential ⟨p/i⟩ is a naive fibration. If p is a +(naive) fibration between fibrant objects then ⟨p/i⟩ is a naive fibration between +fibrant objects hence a fibration. It follows that if i a acyclic cofibration and +j is a cofibration then i ˆ +∼ j is an acyclic cofibration as it is a cofibration by +Definition 2.27 and if p is a fibration between fibrant objects then i ˆ +∼ j has the +right lifting property against p because j has the left lifting property against +⟨p/i⟩. +The case of +→ works exactly the same considering the first half of Re- +mark 2.36. +2.38 Theorem. The category ∞-Catm of m-marked ∞-category admits a left +semi-model structure, called the inductive model structure, in which the cofi- +brations and trivial fibrations are as in Definition 2.27 and the fibrations are as +in Definition 2.33. +Proof. This immediately follows from Theorem 6.12 of [15]. Because of Propo- +sition 2.31 and Lemma 2.37, tensoring by the interval object I of Construc- +tion 2.32 is a “strong Quillen functor” in the sense of section 6 of [15]. Note that +to apply Theorem 6.12 one needs to observe that ∞-Catm, with the (cofibra- +tions, trivial fibrations) and (acyclic cofibrations, fibrations) weak factorization +systems, is both “right saturated” and “left saturated” that is, that a fibration +that has the right lifting property against all cofibrations between cofibrant ob- +jects is a trivial fibration and that a cofibration that has the left lifting property +against all fibrations between fibrant objects is a trivial cofibration. The first +ones hold because the generating cofibrations are cofibrations between cofibrant +objects and the second because that is how we defined acyclic fibrations. +15 + +2.39 Remark. The proof of Theorem 2.38 above also shows that ∞-Catm +also admits a right semi-model category structure whose fibrations and trivial +cofibrations are the fibrations and acyclic cofibrations of Definition 2.33 and +whose cofibrations are as in Definition 2.27. +This however does not clearly make ∞-Catm into a Quillen model struc- +ture but rather into a “two-sided model category” as in Section 5 of [15]. We +refer to Section 5 of [15] for what this means more precisely, but in short, +the problem is that the left and right semi-model categories have different +classes of weak equivalences. The two classes of equivalence however coincide +for arrows that are between fibrant or cofibrant objects. Another way to talk +about this difference is that left and the right semi-model categories are Quillen +equivalent and have the same homotopy category, but define different functors +∞-Catm → Ho(∞-Catm). The two functors agree on objects that are either +fibrant or cofibrant but differ on general objects: one sends an object X to its +cofibrant replacement while the other sends it to a fibrant replacement, and we +do not know if these are always homotopy equivalent when X is neither fibrant +nor cofibrant itself. +2.40 Remark. We do not know if ∞-Catm is actually a Quillen model cate- +gory or not. In the unmarked case, this follows from the fact that all objects +are fibrant. But that is no longer the case in this situation. In terms of the +“two-sided model structure” mentioned in the previous remark, the question is +whether ∞-Catm satisfies one of the equivalent conditions of Proposition 5.3 of +[15]. +We conclude this section with the following lemma that will be useful later: +2.41 Lemma. The map +i+ +n : D♭ +n → (Dn+1, {en+1}) +where en+1 is the unique non-identity arrow of Dn+1, is an anodyne cofibration. +Proof. We will show it is a retract of the map j+ ˆ +∼ in where in is the map +∂Dn → Dn. +In order to achieve this, we will compute j+ ˆ +∼ in more explicitly using the +description of D1 ⊗ Dn given in appendix B.1 of [4] (see proposition B.1.4): As +a polygraph, the generating arrows of D1 ⊗ Dn are the: +a− +0 ⊗ eǫ +k +a+ +0 ⊗ eǫ +k +a ⊗ eǫ +k +where the arrows of D1 have been denoted “a” instead of “e” to distinguish +them, and ǫ is either + or −, k ⩽ n and e+ +n = e− +n . Their source and target are +given as follows: +π−(a− +0 ⊗ eǫ +k) = a− +0 ⊗ e− +k−1 +π+(a− +0 ⊗ eǫ +k) = a− +0 ⊗ e+ +k−1 +π−(a+ +0 ⊗ eǫ +k) = a+ +0 ⊗ e− +k−1 +π+(a+ +0 ⊗ eǫ +k) = a+ +0 ⊗ e+ +k−1 +π−(a ⊗ eǫ +k) = (a− +0 ⊗ eǫ +k)#0(a ⊗ e+ +0 )#1 . . . #k−1(a ⊗ e+ +k−1) +π+(a ⊗ eǫ +k) = (a ⊗ e− +k−1)#k−1 . . . #1(a ⊗ e− +0 )#0(a+ +0 ⊗ eǫ +k) +16 + +We did not put parenthesis in the expression above, to keep them shorter, the +default convention is to do the composition #i in order of increasing values of i. +The last two equations are given by proposition B.1.4 of [4], though note that +this reference is using a different convention than ours regarding the composition +order. Note that the object we are interested in is I +∼ D♭ +n which is the same +polygraph endowed with the special marking where all the arrows a ⊗ eǫ +k are +marked. +We then realize (Dn+1, {en+1}) as a retract of I +∼ D♭ +n+1 as follows: We +call i: (Dn+1, {en+1}) → I +∼ D♭ +n the unique morphism sending en+1 to a ⊗ en. +This is well defined because a ⊗ en is a marked arrow. Next, we define a map +p: I ∼ D♭ +n → (Dn+1, {en+1}) by: +p(aǫ +0 ⊗ eµ +k) = eµ +k if k < n. +p(aǫ +0 ⊗ en) = eǫ +n +p(a ⊗ eǫ +k) = Ieǫ +k if k < n. +p(a ⊗ en) = en+1 +In order to check that this is well defined, we first need to check that this +definition is compatible with the source and target given above, which follow +from an immediate calculation. Then we need to show that this is compatible +with the marking, which is the case as both Ieǫ +k and en+1 are marked. +Finally, the composite p ◦ i send the arrow en+1 to p(a ⊗ en) = en+1 and +hence is the identity of Dn+1. +To conclude the proof, we just have to observe that the maps f and i defined +above send the domain of i+ +n and of j+ ˆ +∼ in to each other. +The domain of j+ ˆ +∼ in is the sub-polygraph of I +∼ D♭ +n which contains all +the generators except a− +0 ⊗ en and a ⊗ en, while the domain of i+ +n contains all +generators of Dn+1 except en+1 and e− +n . +In order to check that the map i is compatible with these sub-polygraphs, +it is enough to check that i(e+ +n ) is in the domain of j+ ˆ +∼ in, to see this, we +compute: +i(e+ +n ) = π+i(en+1) = π+(a ⊗ en) = (a ⊗ e− +n−1)#n−1 . . . #1(a ⊗ e− +0 )#0(a+ +0 ⊗ en) +and we observe that this expression involves neither a− +0 ⊗ en nor a ⊗ en, hence +it does belong to the domain of j+ ˆ +∼ in. +In order to check that the map p is compatible with these sub-polygraphs, we +need to check the image by p of all the generators of I ∼ D♭ +n except a− +0 ⊗ en and +a⊗en. These are given by the formulas p(aǫ +0⊗eµ +k) = eµ +k if k < n, p(a+ +0 ⊗en) = e+ +n +and p(a ⊗ eǫ +k) = Ieǫ +k, which all indeed belong to the image of i+ +n . +3 +Equations and saturations in an m-marked ∞- +category. +The general goal of this section is to arrive at a better description of the fibrant +objects and fibrations between fibrant objects of the model structure of Theo- +rem 2.38. This is achieved using the notion of “equations” in an ∞-categories +introduced by the second named author in [20]. We will recall the basic theory of +equations, in a slightly different language and introduce an analog of equations +to deal with the markings, which we call saturations. +17 + +3.1 +Definitions of equations and saturations +3.1 Definition. A left equation is a special m-marked polygraph (P, M) with +two arrows x, y ∈ P such that: +(1) y is the unique arrow of dimension n + 1 and P contains no arrows of +dimension > n + 1. +(2) y is a marked arrow. +(3) if n ≤ m, x is an unmarked arrow of P. +(4) The source of y admits a decomposition: +π− +n y = ln#n−1(ln−1#n−2...#1(l1#0x#0r1)#1...#n−2rn−1)#n−1rn +where for each i, li and ri are marked i-arrow in P, with ln and rn not +containing x. In particular, x appears only once in π− +n y. +(5) x does not appear in the target of y. +Right equations are defined in the exact same way except the source and +target of y are exchanged in the last two conditions. +We say that (P, M) is an equation to mean that it is either a left or right +equation. If P, with its arrows x and y as in the definition, is an equation one +denotes by ΛP the sub-polygraphs of P that contains all arrow except x and y. +3.2 Remark. Note that specifying the arrows x, y ∈ P is exactly the same as +specifying the subpolygraphs ΛP ⊂ P. For this reason, we will often also call +“equation” the map ΛP → P. +We say that an equation ΛP → P has solutions in C ∈ ∞-Catm if C has the +right lifting property against ΛP → P and we say that a morphism f: C → D +lifts solutions of the equation if it has the right lifting property against the map +ΛP → P. +3.3 Remark. The name “equation” comes from the idea that we are looking for +an element x such that a certain composite of x with other arrows is isomorphic +to another given arrow. From this point of view, a map ΛP → X corresponds +to such an equation in X, and an extension P → X corresponds to a solution +of the equation, or rather the image of x is the solution and y represents the +isomorphism witnessing that x is a solution. +3.4 Definition. A left saturation is a special marked polygraph (P, M) with +arrows x and y satisfying the conditions of Definition 3.1 except that x is a +marked arrow and the target of y is a marked arrow. Right saturations are +defined in the same way. +If P is a saturation, one denotes ΩP the special +m-marked polygraph (P, M − {x}). +3.5 Definition. If P is an equation, we define the m-marked ∞-category +Uni(P) which is the colimit of the following diagram: +∂Dn +D♯ +n +P � +ΛP P +Uni(P) +x∐x′ +z +⌟ +18 + +A map Uni(P) → X corresponds to a map ΛP → X, which is an equation in +X, together with two solutions P → X, given by pairs (x, y) and (x′, y′), and a +marked arrow z: x → x′ which express that the two solutions are isomorphic. +3.6 Definition. Let C be an m-marked ∞-category C and P a left equation +(resp. right equation). +The equation P has solutions in C if for all morphisms ΛP → C, there +exists a lifting (x, y): P → C such that x is sent on a marked arrow whenever +the target of y is (resp. the source of y is). +Solutions to an equation P are C are weakly unique if C has the right lifting +property against P � +ΛP P → Uni(P). +The equation P has unique solutions in C if the equation P has solutions in +C and they are weakly unique. +It will be useful to have a “coherent” version of Uni(P), noted Unicoh(P). +If P is a left equation, P � +ΛP P → Unicoh(P) is obtained as the following +sequence of pushout: +∂Dn +Dn +P � +ΛP P +• +Unicoh(P) +∂Dn+1 +Dn+1 +x∐x′ +z +⌟ +s[x/z]#ny′∐y +⌟ +where s is the source of y. Conversely, if P is a right equation, P � +ΛP P → +Unicoh(P) is obtained as the following sequence of pushout: +∂Dn +Dn +P � +ΛP P +• +Unicoh(P) +∂Dn+1 +Dn+1 +x∐x′ +z +⌟ +y#nt[x/z]∐y′ +⌟ +where t is the source of y. Remarks that in both cases, P � +ΛP P → Unicoh(P) is +an equation. By definition, if C is an m-marked ∞-category such that Unicoh(P) +has a solution in C, then C as the right lifting property against P � +ΛP P → +Uni(P). +3.7 Example. Let n be a non-negative integer. The morphism +j+ ˆ +∼ in: = I ∼ ∂Dn +� +{1} ∼ Dn → I +∼ Dn +is a left equation. Indeed, let y be the top dimensional generator of I +∼ Dn. If +we denote by x the top dimensional arrow of {0} ∼ Dn, and for 0 < k ≤ n, by +ak the image of the top dimensional k-generator of I +∼ Dk−1 by the morphism +I +∼ δ− +k−1: I +∼ Dk−1 → I +∼ Dn, +19 + +Section B.1. of [4] allows to give an explicit description of I +∼ Dn, which we +recalled in the proof of Lemma 2.41. Using this description, we see that if we +name y = a ⊗ en and x = a− +0 ⊗ en the two arrows of I ∼ Dn that are not in the +image of j+ ˆ +∼ in, then we have a decomposition of the source of y of the form: +(((x#0a0)#1a2)...)#n−1an +and all the ak are marked. We denote it +eq +• +• +• +• +n : ΛEq +• +• +• +• +n → Eq +• +• +• +• +n . +3.8 Example. Similarly, the morphism +j+ ˆ +∼ sn: I +∼ Dn +� +{1} ∼ (Dn, {en}) → I +∼ (Dn, {en}) +where sn is the “identity” map Dn → (Dn, {en}) is a left saturation. which +we denote +sat +• +• +• +• +n : ΩSat +• +• +• +• +n → Sat +• +• +• +• +n +3.9 Definition. We define some left equations which play an important role. +In each case, k and n are integers with k ⩽ n. +• eq +• +• +• +k,n : ΛEq +• +• +• +k,n → Eq +• +• +• +k,n , whose target is generated by x and b of di- +mension n, a a marked arrow of dimension k and y: (a#k−1x) ⇒ b. +• eq +• +• +• +k,n : ΛEq +• +• +• +k,n → Eq +• +• +• +k,n , whose target is generated by x and b of di- +mension n, a a marked arrow of dimension k and y: (x#k−1a) ⇒ b. +In all equations above, the domain of the arrow is obtained by removing x +and y. Also, in each case, we have not listed all the constraints of the source +and target that are necessary to make sense of the definition of y. For example, +in Eq +• +• +• +k,n , we have the relation π+ +k−1(a) = π− +k−1(x) for the composition a#k−1x +to exists, and the relations πǫ +n−1(b) = πǫ +n−1(a#k−1x), as b needs to be parallel +to a#k−1x for y to exists. +3.2 +Characterization of fibrant objects +In this section, we will give a simple characterization of the fibrant objects of +the model structure introduced in Theorem 2.38. We will temporarily call the +objects satisfying this characterization “prefibrant” (Definition 3.12) and then +show in Proposition 3.19 that these are exactly the fibrant objects. +3.10 Notation. Suppose given an equation P and a lifting problem of the form: +ΛP +C +P +D +p +Given a a generator of P, we will denote its image in D also by a. If a ∈ ΛP, +we denote by a its image in C. So in general p(a) = a. If the dotted diagonal +lift exists, or in the process of constructing such a lift, the image of x, y ∈ P in +C are also denoted x and y, and we hence also have p(x) = x and p(y) = y. +20 + +3.11 Definition. Let a be a (n + 1)-arrow. An inverse for a is an arrow a−1 +such that there exist two marked arrows: +ǫ: a#na−1 → I +ν: a−1#na → I. +An arrow is invertible if it has an inverse. +3.12 Definition. An m-marked ∞-category C is prefibrant if +(1) marked arrows are invertible and their inverses are marked, +(2) whenever a and c: a → b are marked, so is b . +This directly implies that if b and c: a → b are marked, so is b. +This notion is purely temporary: we will show in Proposition 3.19 that an +object is fibrant for the model structure of Theorem 2.38 if and only if it is +prefibrant. +3.13 Proposition. If C is prefibrant, then equations Eq +• +• +• +k,n and Eq +• +• +• +k,n have +weakly unique solutions in C. +Proof. We show the result by a decreasing induction on k ≤ n. The initialization +corresponds to k = n. +In this case, the data of a morphism ΛEq +• +• +• +n,n → C +corresponds to two n-arrows a and b sharing the same source and such that a is +marked. Let ν: a−1#na → I. If we define x: = a−1#nb and y: ψ#nb: a#nx → b, +the couple (x, y) is a solution of Eq +• +• +• +n,n . If b is marked so is x. We now show +the weak unicity of the solution. Let (¯x, ¯y) be another solution. We then have +a marked arrow: +z: ¯x +ν−1 +−−→ a−1#na#n¯x +¯y−→ a−1 ∗ b. +The assertion for Eq +• +• +• +n,n is similar. +Suppose now the result is true for all k + 1. +We start by showing that +solutions of Eq +• +• +• +k,n and Eq +• +• +• +k,n are weakly unique in C. The data of a morphism +ΛEq +• +• +• +k,n → C corresponds to an n-arrow x: s → t, a k-invertible arrow a, and an +arrow b: a#k−1s → a#kt. Let (x, y: a#k−1x → b) be a solution of this equation. +Let ν: a−1#na → I. The arrow x is then also a solution of Eq +• +• +• +k+1,n: +(ν#0s)#kx = (a−1#k−1a#k−1x)#k(ν#k−1t) +(a−1#k−1b)#k(ν#k−1t) +(a−1#k−1y)#k(ν#k−1t) +and so is weakly unique. The unicity of solution of Eq +• +• +• +k,n is proved similarly. +We show now that Eq +• +• +• +k,n and Eq +• +• +• +k,n have solutions in C. Let (x, y) be a +solution of the equation +(ν#0s)#kx +(a−1#k−1b)#k(ν#k−1t). +y +Moreover, we can find such x marked whenever b is. We then have +(ν#0s)#kx = (a−1#k−1a#k−1x)#k(ν#k−1t). +21 + +By weakly unicity of solution of Eq +• +• +• +k+1,n, we then have a marked arrow +z: a−1#k−1a#k−1x → a−1#k−1b. +But a#k−1x and b are solutions of an equation Eq +• +• +• +k,n , and so there exist a +marked arrow +˜y: a#k−1x → b. +If b is marked, the arrow x that we produce is also marked. The existence of +solution of Eq +• +• +• +k,n is proved similarly. +3.14 Lemma. If equations Eq +• +• +• +k,n +and Eq +• +• +• +k,n +have solutions in C, then all +equations have solutions in C. +Proof. Let P be a left equation. There is a decomposition of the source of y of +the shape +π− +n y = ln#n−1(ln−1#n−2...#1(l1#0x#0r1)#1...#n−2rn−1)#n−1rn +where for each i, li and ri are marked i-arrow in P. We can then use the existence +of solutions to Eq +• +• +• +k,n and Eq +• +• +• +k,n to get two sequences of arrows (xk)0 0) of Defini- +tion 3.25. Indeed, given a weak lifting diagram: +∂Dn +X +Dn +(Dn+1, {en}) +Dn +Y +p +The solid part of the diagram corresponds to a pair of parallel (n − 1)-arrows +(a, b) in X, together with an n-arrow c: p(a) → p(b) in Y , the top dotted mor- +phism gives us an arrow ˜c: a → b, while the bottom dotted morphism corre- +sponds to a marked (n + 1)-arrow e: p(˜c) → c, so this lifting condition corre- +sponds exactly to the third point of Definition 3.25 ( with the second point +corresponding to the case n = 0). +3.5 +The saturated localization. +Proposition 3.19 produces a characterization of fibrant objects of the model +structure of Theorem 2.38: a marked ∞-categories is fibrant if the marked +arrows have inverses and if an arrow isomorphic to a marked arrow is marked. +A careful reader might have noticed however that this is not sufficient to +show that the marked arrows are exactly the arrows that have inverses in the +sense of Definition 3.11. +3.27 Example. Let C be a category, seen as an ∞-category with no non- +identity arrows of dimension > 1. We endow C with the marking C♭, where +only the identity arrows are marked. +With this marking C is fibrant, indeed, it satisfies all the conditions of +Proposition 3.19. +But if the category C has non-identity invertible arrows, +these would be arrows that have inverses in the sense of Definition 3.11 without +being marked. +In this section, we “fix” this problem by introducing a Bousfield localization +in which the fibrant objects have these properties. +3.28 Definition. A marked ∞-category C is said to satisfy the 2-out-of-6 +property if given three composable n-arrow f,g and h such that f#n−1g and +g#n−1h are marked, then f, g and h are marked. +3.29 Remark. If C is a fibrant m-marked ∞-category. +Then the relation +f ∼ g defined by ∃c: f → g a marked (n + 1)-arrow, is an equivalence relation +27 + +on n-arrow. Indeed it is reflexive and transitive as identities are marked and +composites of marked arrows are marked, and it is symmetric as marked arrows +have inverses. +This equivalence relation is moreover compatible with all composition op- +erations, so that one can define a “homotopy n-category” hnC, which is an +n-category whose k arrows for k < n are these of C and its n-arrows are equiva- +lence classes for this relations. We will use in particular that given two parallel +n − 2 arrows u, v in C we have a category hnC(u, v) whose objects are n − 1 +arrows u → v and whose morphisms are equivalence classes of n arrows between +them. +3.30 Lemma. For an m-marked ∞-category C the following conditions are +equivalent: +(1) An arrow in C is marked if and only if it has an inverse in the sense of +Definition 3.11. +(2) C is fibrant in the model structure of Theorem 2.38 and satisfies the 2- +out-of-6 property. +Proof. We first consider C an m-marked ∞-category which satisfies (1), and we +check it is fibrant using Proposition 3.19. The first condition of Proposition 3.19 +is immediate, we check the second condition : if c: a → b is marked and b is +marked, then considering b−1 an inverse of b, the marked arrow connecting +ǫ: b−1#nb → I and ν: b#nb−1 → I, we can simply compose: +b−1#na +b−1#nc +→ +b−1#nb +ǫ→ I +a#nb−1 c#nb−1 +→ +b#nb−1 +ǫ→ I +This shows that b−1 is also an inverse for a, and hence if all arrows with an +inverse are marked a is marked as well. Note that if it is a which is marked in +the first place then one can consider an inverse c−1: b → a and apply the same +argument. +Next, we show that C satisfies 2-out-of-6. +For this, we can rely on Re- +mark 3.29. An n arrow has an inverse in the sense of Definition 3.11 if and +only if it is an isomorphism in the category hnC(u, v) where u and v are its +(n − 2)-dimensional source and target. +Our assumption is then that an n- +arrow is marked if and only if its equivalence class is invertible in the category +hnC(u, v). The fact that marked arrows satisfy 2-out-of-6 then follows from the +fact that isomorphism in a category satisfies the 2-out-of-6 condition. +Conversely, assuming that C satisfies condition (2) we have that marked +arrows have inverses because C is fibrant and Proposition 3.19. If an arrow a +has an inverse a−1 then both a#n−1a−1 and a−1#n−1a are marked because +they are equivalence to identities, and it follows from the 2-out-of-6 condition +that a (and a−1) is marked. +3.31 Theorem. The model structure of Theorem 2.38 admits a Bousfield lo- +calization (as a left semi-model structure) in which the fibrant objects are the +marked ∞-categories which satisfy the equivalent conditions of Lemma 3.30. +We call this model structure the saturated inductive model structure. +28 + +As a Bousfield localization, this model structure has the same cofibrations +and the same fibration between fibrant objects as the model structure from +Theorem 2.38. +Proof. The key point here is that the 2-out-of-6 condition for a marked ∞- +category corresponds to the lifting property against certain cofibrations. +For each n we consider the polygraphs Xn generated by three composable n +arrow +Dn +� +Dn−1 +Dn +� +Dn−1 +Dn +Where each pushout uses the target maps on the left and the source map on +the right. We call f, g and h the three n-dimensional generators of Xn. We +consider the map sn: +sn: +� +Xn, {f#n−1g, g#n−1h} +� +→ +� +Xn, {f, g, h} +� +which is the identity of Xn (with two different markings). sn is a cofibration, +and a marked m-category has the right lifting property against all the sn if and +only if it satisfies the 2-out-of-6 property. +We hence take the left Bousfield localization of the model structure of The- +orem 2.38 at the set sn. The theory of left Bousfield localization for left semi- +model structure can be found in [5] or [15]. +Given that the maps in sn are already cofibration between cofibrant objects, +the fibrant objects of the left Bousfield localization are the fibrant objects that +have the right lifting property against the maps sn and all their iterated cylinder +maps ∇ksn, where if i: A → B is a cofibration, ∇i is the cofibration B � +A B → +IAB for some relative cylinder object. However, in the case of the map sn, given +that it only changes the marking, the pushout B � +A B is just B, hence it is +already a cylinder object, and the map ∇sn is an isomorphism. It hence follows +that an object is fibrant in the localization if it is fibrant and has the lifting +property against all the sn, i.e. has the 2-out-of-6 property. This concludes the +proof. +4 +Comparison with other model structures +4.1 +Truncation functors +4.1 Definition. Let m < p ≤ ∞. There is a functor: +πm: +∞-Catm+1 +→ +∞-Catm +(X, M) +�→ +(X, M). +that marks every arrow of dimension m + 1, an obvious inclusion functor: +ιm+1: +∞-Catm +→ +∞-Catm+1 +(X, M) +�→ +(X, M) +and eventually, a functor: +τm: +∞-Catm+1 +→ +∞-Catm +(X, M) +�→ +(τ(X), M) +29 + +where τ(X) is the sub ∞-category of X whose arrows of dimension strictly su- +perior to m are the ones in M. As M is assumed to be closed under composition +and contains the identities, τ(X) is indeed an ∞-category. +These functors fit in following adjunctions: +πm ⊣ ιm+1 ⊣ τm. +4.2 Notation. For m < p, we also denote by ιp: ∞-Catp → ∞-Catm (resp. +πp: ∞-Catm → ∞-Catp, resp. τp∞-Catm → ∞-Catp) the iterate composite +of ιk (resp. πk, resp. τk) when k range over [p, m]. +Moreover, because ιp is the inclusion of a full subcategory, we will of- +ten identify X and ιpX in our notation. +In the same way, for a morphism +f ∈ Hom(X, τm(Y )), the corresponding morphism in Hom(ιpX, Y ) will also be +denoted f. +4.3 Proposition. For m < p, the adjoint pairs (πm ⊣ ιp) and (ιp ⊣ τm) are +Quillen pairs. +Proof. The functors πm and ιp obviously preserve cofibrations and anodyne +cofibrations. +As mentioned in the introduction, we can consider the two towers of model +structures: +∞-Cat0 π0 +← ∞-Cat1 π1 +← ∞-Cat2 π2 +← . . . +πn−1 +← ∞-Catn πn +← . . . +∞-Cat0 τ0 +← ∞-Cat1 τ1 +← ∞-Cat2 τ2 +← . . . +τn−1 +← ∞-Catn τn +← . . . +and take the projective limit of either tower to get a definition of strict (∞, ∞)- +categories. +Our goal in this section is to show that the inductive model structure on +∞-Cat∞ is equivalent to the limit of the second tower (with τ functors). Here +by projective limit we mean a homotopy theoretic limit of these towers, that is a +homotopy limit of the corresponding tower of (∞, 1)-categories. Such projective +limits of model structures have been studied in [6] and [12] and we will use the +construction from these papers. +4.4 Remark. It should be noted that the results from [6] and [12] are only +proved for Quillen model structures, so they do not immediately apply to the +left semi-model structures that we are using here. The proof from these two +papers easily adapts to the setting of left semi-model structures with very few +modifications, so it should be safe to assume these results can be applied here +as well. Though to avoid relying on this, we will give an independent proof that +the model structure we use as a model of this projective limits exists and state +our main theorem as an equivalence with this model structure. The only aspect +that still relies on applying the results of [6] or [12] to semi-model structures is +in order to interpret our results as saying something about homotopy limits of +towers. +4.5 Definition. A category with weak equivalences is a couple (C, W) where C +is a category and W is a class of map C satisfying the two-out-of-three property. +We define the homotopy category of (C, W) as ho(C, W): = C[W −1]. +30 + +4.6 Definition. We define the category with weak equivalences LimLaxn∈N ∞-Catn, +whose object are sequences X• = {(Xn, fn)}n∈N where Xn ∈ ∞-Catn and +fn: Xn → τnXn+1, and whose weak equivalences are pointwise equivalences. By +adjunction, objects are in bijection with sequences +X0 +f0 +−→ X1 +f1 +−→ . . . +fn−1 +−−−→ Xn +fn +−→ . . . +where each Xn ∈ ∞-Catn. +The category with weak equivalences limn∈N ∞-Catn, is the full sub-category +of LimLaxn∈N ∞-Catn composed of objects {(Xn, fn)}i∈N where for all n, fn: Xn → +τiXn+1 is a weak equivalence of the model structure on ∞-Catn. Weak equiv- +alences are pointwise equivalences. +4.7 Proposition. There exist a model structure on LimLaxn∈N ∞-Catn,, called +the lax-limit structure, where fibrations and weak equivalences are pointwise, and +cofibrations are morphisms h: X• → Y• such that h0: X0 → Y0 is a cofibration +in ∞-Cat0, and for all n, the dotted morphism in the following diagrams is a +cofibration in ∞-Cati+1: +Xn +Xn+1 +Yn +Yn +� +Xn Xn+1 +Yn+1 +Proof. First let us notice that LimLaxn∈N ∞-Catn, can be identified with the +full subcategory of the functors X: N → ∞-Cat∞ such that Xn ∈ ∞-Catn. +There is a model structure on such functors, where fibrations and weak +equivalences are pointwise: the projective (or Reedy) model structure. +The +cofibrations of this model structure are as described in the proposition and this +model structure restricts to LimLaxn∈N ∞-Catn. +4.8 Definition. We have an adjunction +LimLaxn∈N ∞-Catn +∞-Cat∞ +c +τ +⊣ +where the left adjoint sends a sequence X• to its colimit: +c(X•): = Colim +n∈N Xn, +and the right adjoint sends a ∞-marked ∞-category X on the sequence +τ0(X) → · · · → τn(X) → . . . +4.9 Proposition. This adjunction induces a Quillen adjunction between the +lax-limit model structure and the inductive model structure where the left adjoint +preserves weak equivalences and fibrant objects. +31 + +Proof. The left adjoint c clearly preserves cofibrations. Secondly, because the +model structure on ∞-Cat∞ is ω-combinatorial, its fibrant objects are closed +under ω-filtered colimits, and because its factorization systems can be obtained +as ω-accessible functors its weak equivalences are closed under ω-filtered colimit +(this is shown for Quillen model structure as Proposition 7.3 of [11] and for left +semi-model structure as Proposition 7.7 of [13]). This implies that the functor +c preserves cofibrations and weak equivalences (as it is a filtered colimit) and +hence it preserves acyclic cofibrations. +4.10 Proposition. There is a left Bousfield localization of the model structure +on LimLaxn∈N ∞-Catn, called the limit structure, where X• is fibrant if and +only if it is fibrant in the lax-limit model structure and if for all integer n, +fn: Xn → τnXn+1 is a weak equivalence. Moreover, weak equivalences between +fibrant objects are pointwise equivalences. +According to our claim (see Remark 4.4) that the results of [6] or [12] can be +applied to left semi-model structures, the ∞-category obtained as the localiza- +tion of this Bousfield localization is equivalent to the limit of the ∞-categories +obtained as the localization of the ∞-Catn (with the τn functors as transitions). +We need to introduce certain constructions before proving the theorem: +4.11 Construction. Let k be any integer. We define LimLaxi∈k,k+1(∞-Cati, τi) +to be the category whose objects are triplet (X, X′, f: X → τk(X′)) where X and +X′ are respectively k-marked and (k + 1)-marked ∞-categories. By adjunction, +these objects are in bijection with sequences: +X +f−→ X′ +where X and X′ are respectively k-marked and (k + 1)-marked ∞-categories. +There is an adjunction +LimLaxi∈k,k+1(∞-Cati, τi) +LimLaxi∈N(∞-Cati, τi) +U +r +⊣ +where the left adjoint U sends X → Y to the sequence +∅ → ... → ∅ → X +f−→ Y → Y → · · · → Y → . . . +while the right adjoint sends X• to +Xk +f−→ Xk+1. +4.12 Remark. Given i: A ֌ B a cofibration between cofibrant objects in a +(possibly left semi-) model category, we call the (or a) homotopy codiagonal +of i the cofibration B � +A B ֌ IAB where IAB is some choice of a relative +cylinder object for this cofibration. Given that this homotopy codiagonal is +itself a cofibration between cofibrant objects this construction can be iterated. +When constructing a left Bousfield localization at a set S of cofibration between +cofibrant objects, the fibrant objects of the localization are exactly the objects +that have the right lifting property against all arrows in S as well as all their +iterated homotopy codiagonal. +This is a fairly standard result on Bousfield +localization, which is proved for weak model structure (in particular for left +semi-model structure) in [15] (See the proof of Theorem 7.3 and Remark 7.6). +32 + +4.13 Construction. Given A ֌ B a cofibration between cofibrant objects +in ∞-Catk, we can see it as a cofibrant object of LimLaxi∈k,k+1(∞-Cati, τi). +Given a choice of a relative cylinder object IAB for A → B, we have a cofibration +in LimLaxi∈k,k+1(∞-Cati, τi) given by the square: +A +B +B +IAB +A key observation for the proof below is that there is a way to choose a +homotopy codiagonal for this map that is also of this form. +Indeed to construct such a codiagonal map, one needs to construct a (cofi- +bration, weak equivalence) of a map (in the vertical direction): +B � +A B +IAB � +B IAB +B +IAB +One can observe that the horizontal map B � +A B ֌ IAB � +B IAB is already +a relative cylinder object for A ֌ B, so that one can first factorize the leftmost +map +B � +A B +IAB � +B IAB +IAB � +B IAB +B +IAB +∼ +One then forms the pushout P: +B � +A B +IAB � +B IAB +IAB � +B IAB +P +B +IAB +⌜ +And the map P → IAB can then be factored in a cofibration followed by a +weak equivalence. +33 + +B � +A B +IAB � +B IAB +IAB � +B IAB +P +W +B +IAB +⌜ +∼ +Which gives a relative cylinder object, and hence a homotopy codiagonal for +our map of the form: +B � +A B +IAB � +B IAB +IAB � +B IAB +W +But one can see that the object W we constructed above is itself a relative +cylinder object for the map B � +A B → IAB � +B IAB and hence this homotopy +codiagonal is again of the desired form. +Proof of Proposition 4.10. As in Construction 4.13, given a cofibration A ֌ +B in ∞-Catk we consider the cofibration {U(A → B) → U(B → IAB)} in +LimLaxi∈N(∞-Cati, τi). We call Ik the set of cofibrations obtained for A ֌ B +a generating cofibration of ∞-Catk. We claim that a fibrant object (Xi, fi) +of LimLaxi∈N(∞-Cati, τi) has the right lifting property against all maps in Ik +if and only if the map fk: Xk → τkXk+1 is a weak equivalence, and that this +also implies that (Xi, fi) has the right lifting property against all maps of the +form {U(A → B) → U(B → IAB)} when A ֌ B is an arbitrary cofibration in +∞-Catk. +For this, we will use the criterion that in any model category a morphism +between fibrant objects f: X → Y is a weak equivalence if and only if for every +generating cofibration A → B, there is, in the category of arrows, a lifting in all +diagram of shape: +(A → B) +(X +f−→ Y ) +(B → IAB) +where IAB is a relative cylinder object for the cofibration A → B. +This is +proved for weak model categories in Appendix A.2 of [14], see Theorem A.2.6 +and Remark A.2.7. +Now, an object (Xi, fi) of the lax-limit has the right lifting property against +morphisms of Ik if and only if (Xk → Xk+1) has the right lifting property against +(A → B) → (B → IAB). +This last condition is, by adjunction, equivalent +to asking that fk: Xk → τkXk+1 has the right lifting property against (A → +B) → (B → IAB), which is, accorded to the criterium, equivalent to ask that +fk: Xk → τkXk+1 is a weak equivalence. And conversely, if fk: Xk → τkXk+1 is +34 + +an equivalence, then it has the right lifting property against (A → B) → (B → +IAB) for any cofibration A ֌ B and any relative cylinder object. +We then defined the limit model structure as the left Bousfield localization +of the lax-limit model structure by all set Ik (for all values of k). The existence +of this localization is asserted by theorem 7.3 of [15]. +It remains to show that the fibrant object of this localization are the fibrant +object satisfying this condition that fk: Xk → τkXk+1 is a weak equivalence. As +discussed in Remark 4.12, the fibrant object of this localization are the objects +that are fibrant in the lax-limit model structure on which have the left lifting +property against all maps in Ik and all their iterated homotopy codiagonal, but +by Construction 4.13, all these iterated homotopy codiagonals are of the form +U(A → B) → U(B → IAB) and hence, we the discussion above show that their +fibrant objects are exactly the objects such that the map fk: Xk → τkXk+1 is +an equivalence as claimed in the proposition. +4.14 Proposition. The adjunction of Definition 4.8 is a Quillen adjunction +between the limit model structure of Proposition 4.10 and the inductive model +structure. +Proof. We need to show that the adjunction of Proposition 4.14 passes to the +localization, which means that all morphisms of I are sent to trivial cofibrations +in ∞-Cat∞. This is immediate because +U(A → B) → U(B → IAB) +c +�−→ +B → IAB. +4.15 Theorem. The Quillen adjunction between the limit model structure of +Proposition 4.10 and the inductive model structure is a Quillen equivalence. +This induces an equivalence of categories: +ho lim +n∈N ∞-Catn ∼= ho ∞-Cat∞. +Proof. Because the left adjoint preserves all weak equivalences and fibrant ob- +jects, we have to show that for every fibrant ∞-marked ∞-category X, and for +every cofibrant and fibrant sequence X• we have two weak equivalences: +cτX → X +and +X• → τcX•. +The first one is immediate because +X ∼= Colim +n∈N τnX. +Let X• be a cofibrant and fibrant object of the limit model structure. Be- +cause X• and τcX• are fibrant, the second comparison morphism is a weak +equivalence, if and only if for all of k, Xk → Colimn∈N τk(Xn) is a weak equiv- +alence. In order to show this, consider the following diagram: +X0 +... +Xk +Xk +... +Xk +... +X0 +... +Xk +τk(Xk+1) +... +τk(Xn) +... +∼ +∼ +∼ +∼ +∼ +∼ +∼ +∼ +∼ +∼ +∼ +∼ +35 + +where all the vertical morphisms are weak equivalence. Because the left adjoint +c preserves weak equivalence, this induces a weak equivalence: +Xk +∼ +−→ Colim +n∈N τk(Xn) ∼= Colim +n∈N τk(Xn). +4.2 +Comparison with the folk model structure on ∞-Cat +Following [19, Definition 6], we can also give a coinductive definition of invert- +ibility of arrows in an ∞-category. The notion is called “weakly invertible” in +[19]. Explicitly, we define by coinduction: +4.16 Definition. We say that an n-arrow f: a → b in a (marked) ∞-category +is coinductively invertible if there exist g: b → a and two coinductively invertible +(n + 1)-arrows α: f#n−1g → 1a and β: g#n−1f → 1b. +Of course, this implies that there are also (n+1)-arrows α′: 1a → f#n−1g and +β′: 1b → g#n−1f, and then (n + 2)-arrows α#nα′ → 11a , 1f#n−1g → α′#nα, +. . . then followed by several (n + 3)-arrows and so on up to infinity. +4.17 Lemma. Let X be a ∞-category, and M the set of coinductively invertible +arrows. The set M satisfies the two following properties: +(1) M = M. +(2) For all c: a → b in M, a ∈ M ⇔ b ∈ M. +Proof. The first point is the third and the fourth point of example 1.1.9 of [20], +and the second one is a consequence of proposition 1.1.10 of loc cit. +4.18 Proposition. If X is a fibrant m-marked ∞-category, all marked arrows +in X are coinductively invertible in the underlying ∞-category. +Proof. This is a direct consequence of Lemma 3.17. +4.19 Proposition. Let X be an ∞-category and M the set of coinductively in- +vertible arrows. The marked ∞-category (X, M) is then fibrant in the inductive +model structure. +Proof. We show that (X, M) verifies the conditions of Proposition 3.19. +By +definition, marked arrows of (X, M) have inverses in the sense of definition +3.11, and the first condition is fulfilled. +The second condition is implied by +Lemma 4.17. +4.20 Definition. Let G1 be the ∞-category obtained in the factorization of +D1 → D0 in a cofibration k1: D1 → G1 followed by a trivial fibration t1: G1 → +D1 of the folk model structure. +We then define Gn: = Σn−1G1 and kn: = +Σn−1k1: Dn → Gn, tn: = Σn−1t1: Gn → Dn−1. Let us recall that the definition +of the functor Σn−1 is given in Definition 2.4. As the suspension preserves triv- +ial fibrations and cofibrations, the pair (kn, tn) is a factorization of Dn → Dn−1 +into a cofibration followed by a trivial fibration. +36 + +4.21 Proposition. Let X be a ∞-category, and f an n-arrow of X. There +exist a lifting in the following diagram : +Dn +X +Gn +f +if and only if f is weakly invertible. +Proof. This is a reformulation of lemma 18 of [19]. +4.22 Definition. The coinductive model structure is the left Bousfield local- +ization of the model structure on ∞-Cat∞ by the set of morphisms: +{(Gn, ∅) → Dn−1, n ∈ N∗} +4.23 Remark. Remark that if we define ˜ +Gn: = πn−1(Gn, ∅), the sequence +(Gn, ∅) +pn +−→ ˜ +Gn +˜ +kn +−→ Dn−1 +is a factorization in a cofibration followed by a trivial fibration in the inductive +model structure. Using the terminology of [15], we will say that the cofibration +pn represents the morphism (Gn, ∅) → Dn−1. As we can see in the construction +of the left Bousfield localization provides in the proof of the theorem 7.3 of +op cit, a marked ∞-category X is fibrant in the coinductive model structure +if and only if X is fibrant in the inductive model structure and has the right +lifting property against morphisms kn and iterated homotopy codiagonal of kn +for all n > 0. +4.24 Proposition. Let X be a fibrant ∞-marked ∞-category in the inductive +model structure. Then X is fibrant in the coinductive model structure if and only +if marked arrows are exactly the coinductively invertible arrows of the underlying +∞-category. +Proof. Suppose first that X is fibrant in the coinductive model structure and +let f be a coinductively invertible arrow of the underlying ∞-category. +By +proposition 4.21, this corresponds to a morphism f: (Gn, ∅) → X. As remarked +in 4.23, X as the right lifting property against kn, which implies that f can +be lifted by πn−1Gn. That shows that f is marked. Moreover, the proposition +4.18 states that all marked arrows are coinductively invertible. This shows that +marked arrows exactly correspond to coinductively invertible ones. +For the other direction, suppose that X is a marked ∞-category, fibrant +is the inductive model structure, whose marked arrows are the coinductively +invertible ones. We want to show that X is fibrant in the coinductive model +structure. Accorded to Proposition 4.19, X is fibrant is the nonlocalized model +structure. We then have to show for all integers n > 0, X has the left lifting +property against kn and iterated homotopy codiagonal of kn. Remarks now that, +as +˜ +Gn +� +(Gn,∅) ˜ +Gn = +˜ +Gn, all the iterated homotopy codiagonals are identities. +To conclude, it is enough to show that X has the left lifting property against +morphisms kn for n > 0, which is obvious by assumption. +37 + +4.25 Theorem. The subcategory of fibrant objects of the coinductive model +structure on ∞-Cat∞ is isomorphic to ∞-Cat. Moreover, a morphism between +fibrant ∞-marked ∞-categories is a weak equivalence if and only if the corre- +sponding morphism in ∞-Cat is a weak equivalence of the folk model structure. +We then have +hocoind(∞-Cat∞) ∼= hofolk(∞-Cat). +Proof. Let Fib(∞-Cat∞) be the subcategory of fibrant objects of the coinduc- +tive model structure on ∞-Cat∞. We define φ: Fib(∞-Cat∞) → ∞-Cat to be +the functor that forgets the marking. Proposition 4.24 implies that this functor +is an equivalence of category. +Eventually, a morphism f: X → Y between fibrant ∞-marked ∞-categories +is a weak equivalence if and only if, every diagram in the category of arrows of +shape: +(∂Dn → Dn) +(X +f−→ Y ) +(Dn → ˜Gn+1) +admit a lifting. This is equivalent to asking that every diagram in the category +of arrows of ∞-Cat of shape +(∂Dn → Dn) +(X +φ(f) +−−−→ Y ) +(Dn → Gn+1) +admit a lifting, which is equivalent to ask that φ(f) is a weak equivalence. +Note that if m < ∞, then every m-marked ∞-category which is fibrant for +the saturated inductive model structure is also fibrant for the coinductive model +structure, hence when restricting the previous theorem to m-marked objects for +m < ∞, we no longer need to move to the coinductive model structure and we +directly obtain the following: +4.26 Corollary. If m < ∞, the full subcategory of fibrant objects of the satu- +rated inductive model structure on ∞-Catm is isomorphic to the subcategory of +∞-Cat composed of ∞-category whose arrow of dimension strictly superior to +n are coinductively invertible. Moreover, a morphism between fibrant m-marked +∞-categories is a weak equivalence if and only if the corresponding morphism +in ∞-Cat is a weak equivalence of the folk model structure. +4.3 +The folk model structure vs the limit of the π-tower +In this section, we will compare the folk model structure with the limits of the +tower of π functor as considered in Section 4.1. We will show that they are not +equivalent by building a morphism that is not an equivalence of the folk model +structure, but become invertible in the limit of the π-tower. It seems unlikely +38 + +that the limit of the π-tower is actually equivalent to any localization of the +inductive model structure, though we have not been able to give an argument +general enough to show this. +More precisely, we will show: +4.27 Proposition. There exist a morphism f between cofibrant ∞-marked ∞- +category such that +(1) f is not a weak equivalence of the coinductive model structure on ∞- +marked ∞-categories defined in Definition 4.16, +(2) for all integer n, πnf is a weak equivalence of the saturated inductive model +structure on n-marked ∞-categories defined in Theorem 3.31. +As an immediate consequence, we get: +4.28 Corollary. The (∞, 1)-functor from the (∞, 1)-categories associated to +the folk model structure to the limit of the (∞, 1)-categories associated to the +saturated inductive model structure for the n-marked defined in Theorem 3.31, +and induced for all n by the left Quillen functor πn: ∞-Cat∞ → ∞-Catn, is +not an equivalence. +4.29 Construction. Let E1 denote the following 2-polygraphs: +b +b +a +a +f +Ib +Ib +and En: = Σn−1E1. +Let us recall that the definition of the functor Σn−1 is +given in Definition 2.4. When writing Dn → En, we will always consider the +morphism representing the n-arrow Σn−1f. We define by induction a sequence +of polygraphs (Pn)n∈N. We set P0: = D1 and Pn as the pushout: +� +(Pn)n+1 Dn+1 +Pn +� +(Pn)n+1 En+1 +Pn+1 +⌟ +Informally, taking a pushout along Dn+1 → En means freely adding a left +and a right inverse to an arrow f (except there is no marking yet) and so Pn+1 +is constructed by freely adding left and right inverses to all (n + 1)-arrows of +Pn. +When writing D1 → Pn, we will always consider the morphisms representing +the 1-arrow P0 → Pn. Finally, for n ∈ N ∪ {∞} we define Cn and Dn as the +following pushouts: +� +k 0, gn is a (n + k)-generator and g0 = g, +(2) for n > 0, gn appears in the decomposition of the source of gn+1. +Proof. We show this result by coinduction on k. Suppose the result is true for +all (k + 1)-arrows, and let f: a → b be a coinductively invertible k-arrow, and +g a k-generator appearing in the decomposition of f. There exists a k-arrow +f ′: b → a and a coinductively invertible (n + 1)-arrow α: f#k−1f ′ → Ia. As g is +a k-generator appearing in the decomposition f#k−1f ′ (which is the source of +α), we can find a (k + 1)-generator β appearing in the decomposition of α and +such that g is in the decomposition of the source of β. As α is coinductively +invertible, one can continue this process coinductively starting from β to build +a sequence of generators (βn)n∈N satisfying the desired property. We then set +g0: = g, and gn: = βn−1. This sequence also satisfies the desired property. +4.31 Corollary. The ∞-categories C∞ and D∞ have no coinductively invertible +arrow except identities. +Proof. We will show this assertion for C∞, the proof for D∞ is essentially the +same. We proceed by contradiction: let f be a non-identity coinductively in- +vertible k-arrow of C∞. As f is not an identity there should be at least one +k-generator g appearing in its decomposition. As C∞ is a polygraph one can +apply the previous lemma and obtain a sequence (gm)m∈N of generators of C∞. +Eventually shifting the sequence one can freely assume that g0 is of dimension +> 1. The generators of C∞ are obtained by gluing the generators of Pn for all +n at the unique generator of D1, so this g0 will have to be in one of the Pn, it +then follows by induction that all the gm are in the same Pn, but this leads to +a contradiction as the dimension of the generator of Pn is bounded above. +4.32 Corollary. The marked ∞-categories C♭ +∞ and D♭ +∞ are fibrant in the coin- +ductive model structure. +Proof. It is immediate that C♭ +∞ and D♭ +∞ are prefibrant (as in Definition 3.12) +and hence they are fibrant in the indutive model structure. Hence by Propo- +sition 4.24 we only need to check that all their coinductively invertible arrows +are marked, but by the previous corollary, only their identity arrows are coin- +ductively invertible, which concludes the proof. +40 + +4.33 Lemma. The morphism C∞ → D∞ is not a weak equivalence of the +coinductive model structure. +Proof. As both C∞ and D∞ are fibrant in the coinductive model structure, +which is a Bousfield localization of the inductive model structure, this map is a +coinductive equivalence if and only if it is an inductive equivalence. Hence one +can test whether it is an equivalence using Definition 3.25 and Proposition 3.26, +but this map fails to satisfy condition (1) of Definition 3.25, as the 1-arrow of +C∞ corresponding to the vertical map D1 → C∞ is not marked and send to an +identity arrow (hence marked) in D∞. +Let us now show the second point, namely that for any integer n, πnC∞ → +πnD∞ is a weak equivalence. +4.34 Lemma. For all n > 0, the map πn+1En+1 → πnEn+1 is a weak equiv- +alence in saturated inductive model structure for n-marked ∞-category (Theo- +rem 3.31) . +Proof. One should first note that this map is an isomorphism of the underlying +∞-categories and only corresponds to marking all the n-arrows. In particular, it +is a cofibration. Moreover, πn+1En+1 is cofibrant as its underlying ∞-category +is a polygraph. Using the characterization of fibrant objects in the saturated +inductive model structure (see Lemma 3.30 and Theorem 3.31), one easily sees +that fibrant objects have the left lifting property against πn+1En+1 → πnEn+1. +As lifts against these morphisms are unique if they exist, we deduce that any +fibration between fibrant objects has the left lifting property against them. It +follows that this map is an acyclic cofibration and hence a weak equivalence. +4.35 Lemma. For all n, πnPn → D0 is a weak equivalence of the saturated +inductive model structure. +Proof. We define ˜Pn+1 as the pushouts: +� +(Pn)n+1 Dn+1 +Pn +� +(Pn)n+1 πnEn+1 +˜Pn+1 +⌟ +As weak equivalences between cofibrant objects are stable by pushouts, the +previous lemma and the fact that (Dn+1, {en+1}) → πnEn+1 is an acyclic cofi- +bration, imply that all arrows labeled by ∼ in the following diagrams are weak +equivalences: +� +(Pn)n+1 Dn+1 +Pn +� +(Pn)n+1 Dn+1 +Pn+1 +� +(Pn)n+1 πn+1En+1 +πn+1Pn+1 +� +(Pn)n+1(Dn+1, {en}) +πnPn +� +(Pn)n+1 πnEn+1 +˜Pn +� +(Pn)n+1 πnEn+1 +˜Pn +∼ +∼ +∼ +∼ +⌟ +⌟ +⌟ +⌟ +41 + +By two out of three, πn+1Pn+1 → D0 is a weak equivalence if and only if +˜Pn+1 → D0 is, and so if and only if πnPn → D0 is. It remains to show the case +n = 0 which is obvious. +4.36 Lemma. For all n, the induced morphism πnC∞ → πnD∞ is a weak equiv- +alence of the saturated inductive model structure on n-marked ∞-categories. +Proof. Using the last lemma and as weak equivalences between cofibrant objects +are stable by pushout, we have a diagram where all arrows labeled by ∼ are +weak equivalences +� +k∈N πnD1 +� +k∈N πnPk +(� +k0 Xn of simplex of positive dimension called thin simplexes +that includes all degenerate simplexes. +A morphism of m-stratified simplicial sets is a morphism between the under- +lying simplicial sets that sends thin simplexes to thin simplexes. The category +of m-stratified simplicial sets is denoted Strat. +The join is an important operation for simplicial sets, which is defined on +representable by the formula +∆[n] ⋆ ∆[m]: = ∆[n + m + 1] +and extended by colimits to any pair of simplicial set +X ⋆ Y : = +Colim +([n],[m])∈∆×∆ +� +Xn×Ym +∆[n] ⋆ ∆[m]. +See for example [22] Definition 1.2.8.1 and below. We now defined it for stratified +simplicial sets as follows: +42 + +4.38 Definition. If (X, M) and (Y, N) are two stratified simplicial sets, we +define M ⋆ N as the set of simplices of X ⋆ Y of the form x ⋆ y where either x +or y is thin. We then define +(X, M) ⋆ (Y, N): = (X ⋆ Y, M ⋆ N). +4.39 Definition. We define several marking on ∆[n]: +(1) ∆[n]t. The top n-simplex is thin. +(2) ∆k[n]. All simplices that include {k − 1, k, k + 1} ∩ [n] are thin. +(3) (∆k[n])′. All simplices that include {k − 1, k, k + 1} ∩ [n], together with +the (k − 1)-face and the (k + 1) face are thin. +(4) (∆k[n])′′. All simplices that include {k − 1, k, k + 1} ∩ [n], together with +the (k − 1)-face, the k-face and the (k + 1) face are thin. +(5) ∆[3]eq. All simplices of dimension strictly higher than 2, together with +[0, 2] and [1, 3] are thin. +(6) ∆[n]♯. All simplices are thin. +4.40 Definition ([24, Definition 1.19]). An elementary anodyne extension is +one of the following: +(1) The complicial horn inclusions are the regular extensions +Λk[n] → ∆k[n], n ≥ 1, n ≥ k ≥ 0. +(2) The complicial thinness extensions: +(∆k[n])′ → (∆k[n])′′, n ≥ 2, n ≥ k ≥ 0. +(3) The saturation extensions: +∆[n] ⋆ ∆[3]eq → ∆[n] ⋆ ∆[3]♯, n ≥ −1. +(4) The m-triviality extensions: +∆[n] → ∆[n]t, n > m +4.41 Remark. In the case where m = ∞, there is no m-triviality extension. +4.42 Definition. A (saturated) m-complicial set is a marked simplicial set +having the right lifting property against all elementary anodyne extensions. +As demonstrated in [21], m-complicial sets are a model for (∞, m)-categories. +For example, 0-complicial sets and 1-complicial sets are essentially the same +as Kan complexes and quasicategories respectively. The word saturated refers +to the fact that (as in [24]) we have included the “saturation extension” as +part of our elementary anodyne extensions. These are not always included and +play a role similar to the saturated localization of the inductive model structure +considered in Section 3.5. See also [26] for a more general discussion of saturation +for complicial sets. +43 + +4.43 Theorem (Verity [30], Riehl [26], Ozornova-Rovelli [24]). There is a model +structure on Strat where cofibrations are all monomorphisms, and acyclic cofi- +brations are generated by elementary anodyne extension. Fibrant objects of this +structure are the (saturated) m-complicials sets. We denote Stratm the category +Strat endowed with this model structure. +We will use the join to define the adjunction between stratified simplicial +sets and marked ∞-categories. +4.44 Definition. Let C and D be two marked ∞-categories. The joint of C +and D, noted C ⋆ D, is the colimit of the following diagram: +A → {0} → B � A → {1} → B +A → D1 → B +A � B +A ⋆ B +⌟ +As noted in proposition 3.3.11 of [2] at the level of ∞-categories, this is the +usual join of ω-categories, as defined in paragraph 6.30 of [4]. This operation is +then associative. +4.45 Proposition. Let X → Y be a cofibration and K → L an acyclic cofibra- +tion. Morphisms +K ⋆ Y +� +X⋆K +L ⋆ X → L ⋆ Y +and +Y ⋆ K +� +K⋆X +X ⋆ L → Y ⋆ L +are acyclic cofibrations. +Proof. Consider the following diagram: +L � Y +K → ∂D1 → Y ∪ L → ∂D1 → X +K → D1 → Y ∪ L → D1 → X +L � Y +L → ∂D1 → Y +L → D1 → Y +Taking colimit of the lines, this induces a comparison morphism: +K ⋆ Y +� +X⋆K +L ⋆ X → L ⋆ Y. +Proposition 7.5 of [3] implies that this morphism is a cofibration. Lemma 2.37 +implies that vertical morphisms of the previous diagram are weak equivalence. +Furthermore, these colimits are homotopy colimits, the comparison morphism +is then a weak equivalence, and then an acyclic cofibration. We proceed analo- +gously for the second morphism. +4.46 Definition. The terminal category 1 has a monoid structure for this +operation. The multiplication 1 ⋆ 1 → 1 is the unique morphism to the terminal +∞-category. +By the universal property of the category ∆, this induces a cosimplicial +object |−|: ∆ → ∞-Cat∞ where +|∆[n]|: = 1 ⋆ 1 ⋆ ... ⋆ 1. +44 + +The ω-category |∆[n]| is traditionally called the nthoriental. We denote |−|: Sset → +∞-Cat∞ the extension by colimits of this cosimplicial object. For all n, |∆[n]| +is an n-polygraph that admits only one n-generator. If M in a marking for K, +we denote |M| the set of arrows obtained as composition: +Dn → ∆[n] +|v| +−→ K +where the left morphism corresponds to the top cell of the nth orientals, and +the right morphism is in M. We can now extend the realization to stratified +simplicial sets: +|−|: +Strat +→ +∞-Catm +(K, M) +�→ +(|K|, |M|) +This functor is cocontinuous, and induces an adjunction: +Strat +∞-Catm +| | +N +⊣ +The right adjoint is called the stratified Street nerve. By construction, if K and +L are two stratified simplicial sets, we have |K ⋆ L| = |K| ⋆ |L|. +4.47 Remark. In the case m = ∞, this adjunction model the forgetful functor +from strict ∞-categories to weak ∞-categories (given by the stratified Street +nerve N). +The left adjoint corresponds to the “strictification functor” that +sends a weak ∞-category to a strict ∞-category in a universal way. +4.48 Proposition. The stratified nerve preserves fibrant objects. +Proof. Suppose first that m < ∞ and let (X, M) be a fibrant m-marked ∞- +category for the saturated inductive model structure. +According to Corol- +lary 4.26, M consist of coinductively invertible arrows of X, and N((X, M)) +is equal to the stratified simplicial set associated to the Street nerve of X de- +fines in [20, D´efinition 5.2.1]. Theorem 5.2.12 of op.cit. then imply that the +stratified Street nerve sends fibrant objects of the saturated inductive model +structure on ∞-Catm to an m-complicial sets. +Now, let C be a fibrant ∞-marked ∞-category for the saturated inductive +model structure. As the stratified nerve preserves directed colimits, there is an +isomorphism +N(C) ∼= Colim +n∈N N(τnC) +For all n, τnC is fibrant for the saturated inductive model structure for n- +marked ∞-categories, and N(τnC) is then a fibrant of the model structure for n- +complicial sets. As the model structure for ∞-complicial sets is ω-combinatorial, +fibrant objects are stable by directed colimits, and N(C) is fibrant. +4.49 Lemma. The realization functor sends complicial horn inclusion to acyclic +cofibration of the saturated inductive model structure for m-marked ∞-categories. +Proof. The complicial horn inclusion Λ1[2] → ∆[2]1 corresponds to the following +inclusion of marked ∞-categories: +• +• +• +• +• +• +∼ +45 + +which obviously is an equation. The two complicial horn inclusions Λ0[2] → +∆0[2] and Λ2[2] → ∆2[2] are respectively equal to eq +• +• +• +1,1 +and eq +• +• +• +1,1 . +The +realization functor commutes with the join. Furthermore, we can see that for +all 0 < k < n, we have: +∆k[n] = ∆[k − 2] ⋆ ∆1[2] ⋆ ∆[n − k − 2] +and Λk[n] is the sub-object: +∂∆[k − 2] ⋆ ∆1[2] ⋆ ∆[n − k − 2] +∪ +∆[k − 2] ⋆ Λ1[2] ⋆ ∆[n − k − 2] +∪ +∆[k − 2] ⋆ ∆1[2] ⋆ ∂∆[n − k − 2]. +Proposition 4.45 then implies that Λk[n] → ∆k[n] is an equation. We proceed +analogously for the case k = 0 and k = n. +4.50 Theorem. The strictification functor and the stratified Street nerve form +a Quillen adjunction between the model structure for m-complicial sets and the +inductive model structure on ∞-Catm. +Proof. Because of Lemma 4.49, it just remains to show that complicial thinness +extensions, saturation extensions, and m-triviality extensions are sent to acyclic +cofibrations. Let i be such a morphism. According to Proposition 4.48, any +fibrant object of the saturated inductive model structure has the right lifting +property against |i|. As |i| is an identity on the underlying ∞-category, lifts +against it are unique if there exist. This implies that any morphism between +fibrant objects has the right lifting property against |i|, and this morphism is +then an acyclic cofibration. This concludes the proof. +Finally, we can use this to generalize the results from [20]: The stratified +Street nerve: +N: ∞-Cat → sSetm +introduced in [20], is exactly the stratified Street nerve N of the present paper +combined with the fully faithful inclusion ∞-Cat ⊂ ∞-Catm constructed in +Section 4.2, which makes all coinductively invertible arrow marked. Hence: +4.51 Proposition. Let f: X → Y a fibration (resp. a trivial fibration, resp. +weak equivalence) of the canonical model structure on ∞-Cat, then its stratified +Street nerve N(f): N(X) → N(Y ) is a fibration (resp. a trivial fibration, resp. +a weak equivalence) in the Verity model structure on sSetm. +The main result of [20] corresponds to the special case of preservation of +fibrant objects. +Note, that in particular the proposition shows that the stratified Street nerve +from [20], while not being a right Quillen functor, is still a morphism of Brown +categories of fibrant objects, and so it does defines a limit preserving functor on +the corresponding associated ∞-categories. +Proof. As the stratified Street nerve N: ∞-Catm → sSetm is a right Quillen +functor, it preserves fibrations and trivial fibrations, as well as weak equivalences +between fibrant objects. Moreover, we have shown in Section 4.2 that the functor +sending a strict ∞-category to the marked one where the marked arrows are the +coinductively invertible ones, preserves fibrations, trivial fibrations and weak +equivalences. +46 + +References +[1] Fahd Ali Al-Agl, Ronald Brown, and Richard Steiner. Multiple categories: +The equivalence of a globular and a cubical approach. Advances in Math- +ematics, 170:71–118, 2002. +[2] Dimitri Ara. Habilitatation `a diriger des recherche: Th´eorie de l’homotopie +des ∞-cat´egories strictes. 2022. +[3] Dimitri Ara and Maxime Lucas. +The folk model category structure on +strict ω-categories is monoidal. +Theory and Applications of Categories, +35(21):745–808, 2020. +[4] Dimitri Ara and Georges Maltsiniotis. +Join and slices for strict ∞- +categories. ArXiv:1607.00668, 2016. +[5] Michael Batanin and David White. Left bousfield localization without left +properness. ArXiv:2001.03764, 2020. +[6] Julia E Bergner. Homotopy limits of model categories and more general ho- +motopy theories. Bulletin of the London Mathematical Society, 44(2):311– +322, 2012. +[7] Ronald Brown and Philip J Higgins. +The equivalence of ∞-groupoids +and crossed complexes. +Cahiers de topologie et g´eom´etrie diff´erentielle +cat´egoriques, 22(4):371–386, 1981. +[8] Albert Burroni. Higher-dimensional word problems with applications to +equational logic. Theoretical computer science, 115(1):43–62, 1993. +[9] Eugenia Cheng. An ω-category with all duals is an ω-groupoid. Applied +Categorical Structures, 15(4):439–453, 2007. +[10] Sjoerd Erik Crans. On combinatorial models for higher dimensional homo- +topies. Universiteit Utrecht, Faculteit Wiskunde en Informatica, 1995. +[11] Daniel Dugger. Combinatorial model categories have presentations. Ad- +vances in Mathematics, 164(1):177–201, 2001. +[12] Yonatan Harpaz. Lax limits of model categories. Theory and Applications +of Categories, 35(25):959–978, 2020. +[13] Simon Henry. Minimal model structures. Preprint ArXiv:2011.13408, 2020. +[14] Simon Henry. Weak model categories in classical and constructive mathe- +matics. Theory and Applications of Categories, 35(24):875–958, 2020. +[15] Simon Henry. Combinatorial and accessible weak model categories. Journal +of Pure and Applied Algebra, 227(2):107191, 2023. +[16] Chris +Schommer-Pries +(https://mathoverflow.net/users/184/chris- +schommer pries). +Is there an accepted definition of (∞, ∞) category? +MathOverflow. URL:https://mathoverflow.net/q/134099 (version: 2017- +12-15). +47 + +[17] Simon Henry (https://mathoverflow.net/users/22131/simon henry). Test- +ing for equivalences of ∞-categories on strictifications? +MathOverflow. +URL:https://mathoverflow.net/q/313748 (version: 2018-10-25). +[18] Andr´e Joyal and Myles Tierney. Quasi-categories vs segal spaces. Contem- +porary Mathematics, 431(277-326):10, 2007. +[19] Yves Lafont, Fran¸cois M´etayer, and Krzysztof Worytkiewicz. A folk model +structure on omega-cat. Advances in Mathematics, 224(3):1183–1231, 2010. +[20] F´elix Loubaton. +Conditions de kan sur les nerfs des ω-cat´egories. +ArXiv:2102.04281, 2021. +[21] F´elix Loubaton. n-complicial sets as a model of (∞, n)-categories. arXiv +preprint arXiv:2207.08504, 2022. +[22] Jacob Lurie. Higher topos theory. Princeton University Press, 2009. +[23] Fran¸cois M´etayer. Homology, Homotopy and Applications, 10(1):181–203, +2008. +[24] Viktoriya Ozornova and Martina Rovelli. +Model structures for (∞, n)- +categories on (pre) stratified simplicial sets and prestratified simplicial +spaces. Algebraic & Geometric Topology, 20(3):1543–1600, 2020. +[25] A John Power. An n-categorical pasting theorem. In Category theory, pages +326–358. Springer, 1991. +[26] Emily Riehl. Complicial sets, an overture. In 2016 MATRIX Annals, pages +49–76. Springer, 2018. +[27] Richard Steiner. Omega-categories and chain complexes. Homology, Ho- +motopy and Applications, 6(1):175–200, 2004. +[28] Ross Street. Limits indexed by category-valued 2-functors. Journal of Pure +and Applied Algebra, 8(2):149–181, 1976. +[29] Ross Street. The algebra of oriented simplexes. Journal of Pure and Applied +Algebra, 49(3):283–335, 1987. +[30] Dominic RB Verity. Weak complicial sets i. basic homotopy theory. Ad- +vances in Mathematics, 219(4):1081–1149, 2008. +48 + diff --git a/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/load_file.txt b/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b719143ca352d5e7334c75aec0dcd4bc2b333504 --- /dev/null +++ b/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/load_file.txt @@ -0,0 +1,1749 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf,len=1748 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='11424v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='CT] 26 Jan 2023 An inductive model structure for strict ∞-categories Simon Henry and Felix Loubaton Abstract We construct a left semi-model category of “marked strict ∞-categories” for which the fibrant objects are those whose marked arrows satisfy nat- ural closure properties and are weakly invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The canonical model structure on strict ∞-categories can be recovered as a left Bousfield local- ization of this model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We show that an appropriate extension of the Street nerve to the marked setting produces a Quillen adjunction between our model category and the Verity model structure for complicial sets, generalizing previous results by the second named author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Finally, we use this model structure to study, in the setting of strict ∞-categories, the idea that there are several non-equivalent notions of weak (∞, ∞)- categories - depending on what tower of (∞, n)-categories is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We show that there ought to be at least three different notions of (∞, ∞)- categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Contents 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 The street nerve as a right Quillen functor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 The two (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=') notions of (∞, ∞)-categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3 2 ∞-categories and marked ∞-categories 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 ∞-categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Marked ∞-categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 Tensor product of m-marked ∞-categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 27 4 Comparison with other model structures 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 Truncation functors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Comparison with the folk model structure on ∞-Cat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 The folk model structure vs the limit of the π-tower .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 Complicial sets and stratified Street nerve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 42 1 1 Introduction In the present paper, we introduce (in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2) a category ∞-Catm of “m- marked (strict) ∞-categories” for m ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The objects of ∞-Catm are strict ∞-categories, with, similarly to stratified simplicial sets, some arrows be- ing “marked”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The marked arrows are required to be closed under composition, and all identities arrows as well as all arrows of dimension > m are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This category ∞-Catm is equipped with two monoidal closed structures denoted → and ∼ that are both the Gray-Crans tensor product on the underlying strict ∞-categories but act differently on markings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' These two monoidal structures are meant to respectively be models for the“lax-Gray tensor product” and the “pseudo-Gray tensor product”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Our main result is the construction of a model structure1 on ∞-Catm similar to the canonical (or “Folk”) model structure on strict ∞-category from [19]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' There is a combinatorial left semi-model structure on the cate- gory ∞-Catm of m-marked ∞-categories such that: This model structure is monoidal for both tensor products ∼ and → (from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The cofibrations are the map that are cofibrations of the canonical model structure between the underlying ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The fibrant objects are the marked ∞-categories in which all marked arrows admit marked weak inverses, and in which if there is a marked arrow a → b then a is marked if and only if b is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Fibrations between fibrant objects are the “isofibrations” (as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Weak equivalences between fibrant objects are “equivalence of marked ∞- categories”(as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This model structure is a model for strict “(∞, m)-categories” where “invert- ibility” or arrows of dimension > m is taken in a weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The existence of this model structure is established in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4, but some of its properties, in particular, the characterization of fibrant objects and fibrations between fibrant objects will only be established in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We also consider two left Bousfield localizations of this model structure: The saturated inductive model structure, studied in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='5, whose fibrant objects are the ∞-categories in which every arrow which is weakly invertible up to marked arrows is also marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The coinductive model structure, studied in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2, whose fibrant objects are the ∞-categories in which every coinductively invertible (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='16) arrow is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This second localization is equivalent2 to the canonical model structure on ∞-categories from [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The motivations to introduce this model structure come from two different lines of investigations that we will explain separately below: 1We use the term “model category” as a generic name for all sorts of model categories (Quillen model categories, semi-model categories, weak model categories, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=') 2Though not through a Quillen equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 The street nerve as a right Quillen functor In [20], the second named author has shown that the Street nerve of a strict ∞-category can be made into a complicial set by defining the “thin” simplexes as being those whose top dimensional arrows are weakly invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' From there, it is natural to ask whether this stratified version of the Street nerve, also preserves fibrations, and hence is a morphism of categories of fibrant objects (and this will be shown in the present paper as Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In fact, more generally, one could ask if it is possible to make this version of the Street nerve into a right Quillen functor (for the Verity model structure on complicial sets from [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is not directly possible simply because this stratified Street nerve is not a right adjoint functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The solution to this problem is to work with marking on both sides: The usual Street nerve from strict ∞- categories to simplicial sets is a right adjoint functor, and one can extend it to a right adjoint functor from marked ∞-categories to “marked” simplicial sets (or rather stratified simplicial sets to follow the terminology of [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 we show that this functor is indeed a right Quillen functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This right Quillen functor from marked ∞-categories to stratified simplicial sets is meant to be a model for the forgetful functor from strict (∞, ∞)-categories to weak (∞, ∞)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In particular, the corresponding left Quillen functor from stratified simplicial sets to marked ∞-categories is a model for the more mysterious “strictification functor”, sending weak ∞-categories to strict ∞- categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' At the level of ∞-groupoids, this strictification functor corresponds essen- tially to (non-abelian) homology, through the equivalence between strict ∞- groupoids and crossed chain complexes ([7]) which is well-known to be a conser- vative functor by Whitehead’s theorem for homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The first named author has conjectured [17] that more generally this strictification functor should be conservative on weak (∞, m)-categories for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This allows us to state a concrete version of this conjecture here: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The left Quillen functor | |: sSetm → ∞-Catm from Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 reflects weak equivalence between cofibrant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 The two (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=') notions of (∞, ∞)-categories C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='Schommer-Pries and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='Rezk have independently argued ([16]) that there should be more than one notion of weak (∞, ∞)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' More precisely, they both arrive at the conclusion that even if one accepts (which seems to be a clear consensus nowadays) that there is only one notion of weak (∞, n)- categories for finite n, there are at least two different ways to build a notion of (∞, ∞)-categories out of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Before we go into further details, we should say that the following discussion is mostly informal and speculative and most of it has not been formalized in any models - in fact, one motivation for the present paper is to formalize some of it in the context of strict ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' First, let us go over the argument put forward by Rezk and Schommer- Pries, or at least how we understand it: The forgetful (or inclusion) functor from (∞, n)-categories to (∞, n + 1)-categories is supposed to have both a left adjoint πn, which freely adds inverses to all (n + 1)-arrows and a right adjoint τn which remove all non-invertible (n + 1)-arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3 This allows to produce two different towers: (∞, 0)-Cat π0 ← (∞, 1)-Cat π1 ← (∞, 2)-Cat π2 ← .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' πn−1 ← (∞, n)-Cat πn ← .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (∞, 0)-Cat τ0 ← (∞, 1)-Cat τ1 ← (∞, 2)-Cat τ2 ← .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' τn−1 ← (∞, n)-Cat τn ← .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' and one can take the projective limit of either of these two towers to give a definition of what is an (∞, ∞)-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If one takes the limits of the π-tower then one can see that an arrow that is “coinductively” invertible (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='16) has to be considered invert- ible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' To be precise, we mean that if F: X → Y is a morphism in the limit of the π-tower which admits an inverse up to a coinductively invertible natural transformation then F is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The situation in the limit of the τ-tower however is fairly different: Given an (∞, ∞)-category in this sense, it corresponds to a collection of (∞, n)-categories Xn such that Xn ≃ τnXn+1, and an n-arrow corresponds to an n-arrow of Xn (or of Xk for k > n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In this setting one has an intrinsic notion of equivalence: an n-arrow is said to be an equivalence if it belongs to Xn−1 (equivalently if it is invertible in the (∞, n)-category Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In this setting, coinductively invertible arrows do not have to be invertible if none of the higher cells witnessing the coinductive invertibility are not themselves invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' To clearly show that the two are different, one can for example consider the (∞, ∞)-category of cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In the limit of the τ-tower one can define it by taking Xn to be the (∞, n)-categories of cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In this (∞, ∞)-category, every arrow has a dual, so it follows from a result of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='Cheng (see [9]) that every arrow in the cobordisms (∞, ∞)-category is coinductively invertible, although there are many non-invertible n-arrows in Xn for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Hence, if one were trying to define Xn in the limit of the π-tower, it would be equivalent to an ∞-groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Using our model structure of marked strict ∞-category, we will make these two constructions formal in the context of strict ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is of course only meant to be a toy model for the case of weak ∞-categories, but it is already interesting, and it will show that the picture above while correct, needs to be refined a little.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' First, we will show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 that our model structure on ∞-Catm for m = ∞ corresponds to the limit of τ-tower as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' More precisely, we will show that it is Quillen equivalent to an appropriate homotopy limit of the ∞-Catm for m < ∞ using the τn functor as transition functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The notion of homotopy limit of a tower of model structure we are using has been introduced in [6], and we will use their construction of the homotopy limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Here there is a small gap we should disclaim: [6] only develops the theory of such limits for Quillen model categories and not semi-model categories, and we will apply their construction to our left semi-model categories directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In order for our argument to be complete despite this, we will prove that the construction from [6] does yield to a left semi-model category, but we will not reprove that it corresponds to a homotopy limit as in [6, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' However, it should be noted that in practice, the argument of [6] seems to carry over to our setting with almost no changes, so this gap is not really a concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 4 In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 we will show that the folk model structure is equivalent to the left Bousfield localization of our model structure which corresponds to turning all coinductively invertible arrows into equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' However, we will also show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3, that the folk model structure is not equivalent to the limit of the π-tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It is unclear if the limit of the tower of πn corresponds to further localization of our model structure, or is something entirely different, but we find that the argument we will give in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 to distinguish between the folk model structure and the limit of π-tower shows that this limit is exhibiting behaviors that are not really expected from a notion of (∞, ∞)-categories, or at least are not typical of any known model of ∞- categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Coming back to the world of weak (∞, ∞)-categories, this suggests that the two most interesting notions of weak (∞, ∞)-categories are the limit of τn tower, which corresponds to an “inductive” notion of equivalences, and its localization that turn the coinductive equivalence into equivalences, but this localization should be different from the limit of the πn-tower which might not be an interesting notion of (∞, ∞)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' What we mean here is that we are not aware of any attempt of giving a concrete definition of (∞, ∞)-categories that seems to produce something that could be equivalent to this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' All definitions we have seen can be reasonably conjectured to be equivalent to either the limit of the τn tower or to its “coinductive” localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2 ∞-categories and marked ∞-categories 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 ∞-categories A globular set is a presheaf on the globular category G: D0 D1 D2 D3 D4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' i+ 0 i− 0 i+ 1 i− 1 i+ 2 i− 2 i+ 3 i− 3 With the relations iǫ ni+ n−1 = iǫ ni− n−1 for all n > 0 and ǫ ∈ {+, −}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We also denote by iǫ k the map Dk → Dn for k < n obtained by composing any string of arrow ending with iǫ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' These and the identity arrows are the only maps in the category G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If X is a globular set, one denotes by Xn the set X(Dn) whose elements are called n-arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The map Xn → Xk induced by iǫ k: Dk → Dn is denoted by πǫ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An ∞-category is a globular set X together with operations of compositions Xn ×Xk Xn → Xn (0 ≤ k < n) which associates to two n-arrows (x, y) verifying π+ k (x) = π− k (y), one n-arrow x#ky, as well as identities Xn → Xn+1 associating to an n-arrow x, an (n + 1)-arrow Ix, and satisfying the following axioms: 5 (1) ∀x ∈ Xn, πǫ n(Ix) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (2) π− k (x#ny) = π− k (x) and π+ k (x#ny) = π+ k (y) whenever the composition is defined and k ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (3) πǫ k(x#ny) = πǫ k(x)#nπǫ k(y) whenever the composition is defined and k > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (4) x#nIπ+ n x = x and Iπ− n x#nx = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (5) (x#ny)#nz = x#n(y#nz) as soon as one of these is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (6) If k < n (x#ny)#k(z#nw) = (x#kz)#n(y#kw) when the left-hand side is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A morphism of ∞-categories is a map of globular sets commuting with both operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The category of ∞-categories is denoted ∞-Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An (n + 1)-arrow c in an ∞-category is said to be trivial, or an identity arrow, if there exists an n-cell d such that c = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' By abuse of notation, we also denote Dn the ∞-category that admits for any k < n only two k-non-trivial arrows, denoted e− k and e+ k , and a single non-trivial n-arrow, denoted en verifying : π− l (eǫ k) = e− l π+ l (eǫ k) = e+ l for l ≤ k < n π− l (en) = e− l π+ l (en) = e+ l for l ≤ n The ∞-category ∂Dn is obtained from Dn by removing the n-arrow en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We thus have a morphism in: ∂Dn → Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that ∂D0 = ∅ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If X is an ∞-category, we define the globular set ΣX, called the suspension of X, by the formula (ΣX)0 = {a, b} (ΣX)n+1: = Xn ∪ {Ina, Inb} where In a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In b ) is the n-times iterated unity of a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' of b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Moreover, ΣX inherits from X a structure of ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Eventually, for an integer n, we define the ∞-category ΣnX, called the n- suspension of X, as the n-times iterated suspension of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Next, we define the notion of polygraphs, first introduced under the name “computads” by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Street in [28] for 2-categories, with the general notion being hinted at in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' As far as we know the first formal introduction of polygraphs in the literature is in [25] and independently in [8], where the name “polygraphs” was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Here we will exploit that the category of polygraphs identifies with a (non-full) subcategory of ∞-Cat to give a shorter definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We refer to the references above for a more complete introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='5 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that an ∞-category X is a polygraph if it can be constructed from the empty ∞-category by freely adding arrows with specified source and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' That is if X can be obtained as a transfinite composition ∅ = X0 → X1 → · · · → Xi → Colim Xi = X where for each i, the map Xi → Xi+1 is a pushout of Y × ∂Dn → Y × Dn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An arrow of a polygraph is said to be a generator if it is one of the arrows that has been freely added at some stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A morphism of ∞-categories between two polygraphs is said to be a mor- phism of polygraphs or a polygraphic morphism if it sends each generator to a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An n-polygraph is a polygraph whose generators are all of dimension ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='6 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Generators of a polygraph can be shown to be exactly the arrows that cannot be written as a composite in a non-trivial way, so the notion of generator does not depend on the choice of the presentation of X, and any isomorphism between polygraphs is automatically polygraphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='7 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The only n-polygraphs for n < 0 is the empty ∞-category, the category of 0-polygraphs is equivalent to the category of sets and corresponds to discrete ∞-categories, the category of 1-polygraphs (and polygraphic mor- phisms between them) is equivalent to the category of directed graphs, and they corresponds to categories that are free on a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will sometimes distinguish between a polygraph seen as an object of the category of polygraphs and polygraphic morphisms, and the corresponding ∞-category, which we call the free ∞-category on the polygraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='8 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Each arrow in a polygraph can be written as an iterated compos- ite of the generators (not necessarily in a unique way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For an n-arrow f, the set of generators of dimension n that appear in such an expression (and even the number of times they appear) is the same for all such expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will say that an n-generator appears in an n-arrow if it appears in any such expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='9 Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The category ∞-Cat admits a closed monoidal structure, called the Gray tensor product or Crans-Gray tensor product, which we denote as ∞-Cat × ∞-Cat → ∞-Cat X, Y �→ X ⊗ Y Its explicit construction is very involved and we will assume the reader is already familiar with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It was first introduced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Crans in his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' thesis [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We refer to [1] for an introduction to this tensor product close to its original definition, and to [27] for a more modern account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The proof of the existence of this monoidal structure in [27] contains some gaps that have been fixed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It is easy to see from either of these definitions that Dn ⊗ Dm has a unique non-trivial arrow of dimension n + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If f and g are respectively an n-arrow of X and an m-arrow of Y , which corresponds to morphisms f: Dn → X and g: Dm → Y , we denote by f ⊗ g the m + n arrow of X ⊗ Y obtained as the image of this non-trivial (n+m)-arrow by the functor f ⊗g: Dn ⊗Dm → X ⊗Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We recall from [3]: 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='10 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If X and Y are polygraphs then X ⊗ Y is also a polygraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The generators of X ⊗ Y are the arrow of the form x ⊗ y where x and y are respectively generators of X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Finally, we recall from [19] that ∞-Cat carries a model structure, called the folk model structure in which every object is fibrant and where the generating cofibrations are the ∂Dn → Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Its weak equivalences are a natural class of equivalence of ∞-categories that generalizes the equivalences of ordinary cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It was shown in [23] that the cofibrant objects are exactly the polygraphs and it also follows from this that the cofibrations between cofibrant objects are the polygraphic inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It was shown in [3] that this model structure is a monoidal model structure for the Gray tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Marked ∞-categories For the rest of the article, we fix an m ∈ N ∪ {∞} 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='11 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An m-marked ∞-category is an ∞-category X, together with a set M ⊂ � k>0 X(k) of arrows of positive dimension called marked arrows such that: All identity arrows Ix are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' All arrows of dimension strictly superior to m are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If x and y are marked n-arrows and x#ky is defined, then x#ky is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A morphism of m-marked ∞-categories is a morphism between the under- lying ∞-categories that sends marked arrows to marked arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The category of m-marked ∞-categories is denoted ∞-Catm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that if m = ∞, then the second condition of the definition simply disappears, this is the main case we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='12 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If X is an ∞-category we denote by X# the m-marked ∞- category (X, X>0) where all arrows of positive dimension are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We denote by X♭ the m-marked ∞-category where only identity arrows and k-arrows for k > m are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='13 Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If X is an ∞-category and M ⊂ � k>0 Xk is a set of arrows of X, we denote by M the smallest set of arrows such that M ⊂ M and (X, M) is an m-marked ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' That is M is the reunion of the set of arrows of dimension strictly superior to m and the set of all n-arrows that can be written as iterated composites of n-arrows in M and arrows of the form Ix for x an (n − 1)-arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For example X♭ = (X, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='14 Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The category of m-marked ∞-categories has all colimits, and they are easily described in terms of colimits of ∞-category and of Con- struction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='13: if (Xi, Mi)i∈I is a diagram of m-marked ∞-category indexed by a category I then: Colim i∈I (Xi, Mi) = � Colim i∈I Xi, ∪ifi(Mi) � where fi denotes the canonical map fi: Xi → Colimi∈I Xi and fi(Mi) is simply the set of arrows of the form fi(x) for x ∈ Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 8 This is easily shown by checking that the right-hand side has the universal property of the colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='15 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A special m-marked polygraph is an m-marked ∞-category of the form (X, M) where X is free on a polygraph and M only contains generators of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='16 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If (X, M) is a special m-marked polygraph, then an n-arrow f is in M if and only if n > m or if all the generating n-arrows that appear in f are in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An arrow satisfying this condition is a composite of marked n-arrows and identities of lower dimensional arrows, so it has to be in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Conversely, this set of arrows contains M and all identities (as no n-dimensional arrows appear in their expression) and is closed under composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 Tensor product of m-marked ∞-categories In this section we construct two monoidal closed structures on the category of m-marked ∞-categories, respectively called the pseudo-Gray tensor product ∼ and the lax-Gray tensor product →.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Both are obtained by putting different markings on the Gray tensor product from Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For example, the lax-Gray tensor product D1 → D1 is C♭ 1 where C1 is the polygraph C1 = \uf8eb \uf8ec \uf8ed \uf8f6 \uf8f7 \uf8f8 while D1 ∼ D1 is the special m-marked polygraph (C1, D) where D only contains the unique 2 dimension generator of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' So, unless m = 0 or m = 1, the two tensor products are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' At the derived or homotopy theoretic level, the pseudo-Gray tensor product should correspond to the cartesian product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The formal definition goes as follows 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='17 Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Given two m-marked ∞-categories (X, M) and (Y, N) we define two sets of arrows in X ⊗ Y : M → N is the set of arrows of the form x ⊗ y ∈ X ⊗ Y where either x ∈ M or y ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' M ∼ N contains all arrows in M → N together with all arrows of the form x ⊗ y with x and y both of dimension > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that M → N and M ∼ N are not marking on X ⊗Y : they are not stable under composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' So we define: (X, M) → (Y, N) = (X ⊗ Y, M → N) (X, M) ∼ (Y, N) = (X ⊗ Y, M ∼ N) We will show in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='37 that both make the category of m-marked ∞-categories into a monoidal closed category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In order to show this, it is convenient to introduce the following notations: 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='18 Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For A and B subsets of arrows in ∞-categories, we denote by A ⊗ B the set of arrows of the form a ⊗ b ∈ X ⊗ Y for a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For X and ∞-category, we denote by X⩾0 the set of all arrows of X and by X>0 the set of all arrows of dimension > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We can hence, for (X, M) and (Y, N) to m-marked ∞-category rewrite the definitions above as: M → N = (M ⊗ Y⩾0) ∪ (X⩾0 ⊗ N) M ∼ N = (M → N) ∪ (X>0 ⊗ Y>0) = (M ⊗ Y⩾0) ∪ (X⩾0 ⊗ N) ∪ (X>0 ⊗ Y>0) By definition of the Gray tensor product, we have the following result: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='19 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X and Y be two ∞-categories, then X⩾0 ⊗ Y⩾0 = (X ⊗ Y )⩾0 X>0 ⊗ Y⩾0 ∪ X⩾0 ⊗ Y>0 = (X ⊗ Y )>0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' That is X ⊗ Y is generated under composition by arrows of the form x ⊗ y, and the arrows of dimension > 0 of X ⊗ Y are generated under compositions by arrows of the form x ⊗ y with x or y of dimension > 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='20 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X be an ∞-category and M, N two subsets of arrows of X then: M ∪ N = M ∪ N = M ∪ N = M ∪ N Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='21 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X, Y be two ∞-categories and M ⊂ X⩾0 and N ⊂ Y⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Then: M ⊗ N = M ⊗ N = M ⊗ N = M ⊗ N Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will only show the equality M ⊗ N = M ⊗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The equality M ⊗ N = M ⊗ N is proved in the exact same way and the last equality follows immedi- ately by applying the result to M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will also only proves the results for m = ∞, the case of a general m follows immediately as it marks all arrow of dimension > m on each side of these equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The evident inclusion M ⊂ M implies M ⊗ N ⊂ M ⊗ N, so it is then enough to show that M ⊗ N ⊂ M ⊗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let K be the set of arrows k in X such that k ⊗ n ∈ M ⊗ N for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We need to show that K is closed by identity and composition to finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If k = Ix, then k ⊗ n = Ix⊗n ∈ M ⊗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let now k, k′ ∈ K of dimension n such that k#ik′ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' They are encoded by a map Dn � Di Dn → X and let y ∈ N be an arrow of dimension m of Y , encoded by a map Dm → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Together these induced a map e: � Dn � Di Dn � ⊗Dm → X⊗Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' � Dn � Di Dn � ⊗ Dm is a polygraph of dimension m + n with only two generating arrows of maximal dimensions that are sent to k ⊗ y and k′ ⊗ y, which are by hypothesis in M ⊗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Now the arrow corresponding to (k#ik′) ⊗ y in � Dn � Di Dn � ⊗ Dm is in M ⊗ N as all the top dimensional generators that appear in it are in M ⊗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We have proved that k#ik′ ⊗ y ∈ M ⊗ N for all y ∈ N, hence k#ik′ ∈ K and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='22 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X, Y be two ∞-categories, M ⊂ X⩾0 and N ⊂ Y⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Then we have M → N = M → N M ∼ N = M ∼ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Given the formula for M → N and M ∼ N from Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='18, this is a direct consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='20 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='23 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X, Y, Z be three ∞-categories, M ⊂ X>0, N ⊂ Y>0 and P ⊂ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Then we have (M → N) → P = M → (N → P) (M ∼ N) ∼ P = M ∼ (N ∼ P) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We begin with the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let E: = (M ⊗ Y⩾0 ⊗ Z⩾0) ∪ (X⩾0 ⊗ N ⊗ Z⩾0) ∪ (X⩾0 ⊗ Y⩾0 ⊗ P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='19, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='20 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='21 implies the following equalities: E = M ⊗ Y⩾0 ⊗ Z⩾0 ∪ X⩾0 ⊗ (N ⊗ Z⩾0 ∪ Y⩾0 ⊗ P) = M ⊗ (Y ⊗ Z)⩾0 ∪ X⩾0 ⊗ (N → P) = M → (N → P) A very similar computation also shows that E = (M → N) → P, which concludes the proof of the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For the second equality, we define F: = (X⩾0 ⊗ Y>0 ⊗ Z>0) ∪ (X>0 ⊗ Y⩾0 ⊗ Z>0) ∪ (X>0 ⊗ Y>0 ⊗ Z⩾0) The second equality of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='19 implies that: F = Xk⩾0 ⊗ Y>0 ⊗ Z>0 ∪ X>0 ⊗ (Y ⊗ Z)>0 and then that E ∪ F = M ⊗ (Y ⊗ Z)⩾0 ∪ X⩾0 ⊗ (N ∼ P) ∪ X>0 ⊗ (Y ⊗ Z)>0 = M ∼ (N ∼ P) and here again, a similar computation shows E ∪ F = (M ∼ N) ∼ P, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='24 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X be an ∞-category, M ⊂ X>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Then the empty set, considered as a subset of the ∞-category D0, verifies (up to the identifications D0 ⊗ X ≃ X ⊗ D0 ≃ X): ∅ → M = M → ∅ = M ∅ ∼ M = M ∼ ∅ = M Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The first equality is a straightforward application of the definition of →.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For the second case, we also use that all arrows of (D0)>0 ⊗ X>0 are identities and so all belong to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='25 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Both the lax-Gray tensor product → and the pseudo-Gray tensor product ∼ as defined above are monoidal structures on the category of m-marked ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In both cases the forgetful functor to ∞-categories is monoidal and their unit is D♭ 0 = D# 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that D♭ 0 = D# 0 = (D0, ∅) as all arrows of D0 of dimension > 0 are identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The proposition exactly says that the structural map (associativity and unit isomorphism) of the Gray tensor product of ∞-categories preserves the marking we specified on the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For the unit, let (X, M) be an m-marked ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='21 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='24 imply that (X, M) → (D0, ∅) = (X ⊗ D0, M → ∅) = (X, M) (X, M) ∼ (D0, ∅) = (X ⊗ D0, M ∼ ∅) = (X, M) and (D0, ∅) → (X, M) = (D0 ⊗ X, ∅ → M) = (X, M) (D0, ∅) ∼ (X, M) = (D0 ⊗ X, ∅ ∼ M) = (X, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For the associativity isomorphism, let (X, M), (Y, N) and (Z, P) be three ∞- categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='21 implies that � (X, M) → (Y, N) � → (Z, P) = (X ⊗ Y ⊗ Z, (M → N) → P) � (X, M) ∼ (Y, N) � → (Z, P) = (X ⊗ Y ⊗ Z, (M ∼ N) → P) and (X, M) → � (Y, N) → (Z, P) � = (X ⊗ Y ⊗ Z, M → (N → P)) (X, M) ∼ � (Y, N) → (Z, P) � = (X ⊗ Y ⊗ Z, M ∼ (N → P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='23 shows that these two marking on X ⊗ Y ⊗ Z, in the lax and the pseudo case, coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='26 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The pseudo and lax-Gray tensor product → and ∼ preserves colimits in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In particular, as ∞-Catm is locally presentable, this immediately implies that both tensor products are closed monoidal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It follows from the fact that the Gray tensor product ⊗ preserves colimits in each variables, the description of colimits of m-marked ∞-category given in Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='14 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 The semi-model structure In this section, we will construct a left semi-model structure on the category ∞-Catm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='27 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We define the set I = Im ∪ Ia to be our set of generating cofibrations in ∞-Catm where: Ia = {in: ∂Dn → Dn, |n ⩾ 0} 12 Im = {Dn → (Dn, {en}) , n ⩾ 0} An arrow in ∞-Catm is said to be a trivial fibration if it has the right lifting property against all arrows in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An arrow in ∞-Catm is said to be a cofibration if it has the left lifting property against all trivial fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='28 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It immediately follows from the small object argument that every arrow can be factored into a cofibration followed by a trivial fibration and that all cofibrations are retracts of transfinite compositions of pushouts of arrows in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='29 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An arrow π: X → Y has the right lifting property against all arrows in Ia if its image by the forgetful functor to ∞-Cat is a trivial fibration, that is if for every pair of parallel n-arrows u, v in X, the map HomX(u, v) → HomY (π(u), π(v)) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' π has the right lifting property against all arrows in Im if and only for every arrow f ∈ X such that π(f) is marked in Y , f is marked in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A trivial fibration is a map that has both these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='30 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The cofibrant objects of ∞-Catm are exactly the m-marked ∞- categories whose underlying ∞-category is free on a polygraph, with any possible marking on them (not just the special markings of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Indeed, transfinite compositions of pushouts by arrows in Ia only starting from the empty ∞- category exactly give all polygraphs with no markings on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Pushouts by Im are simply changing the marking and can make any arrow marked, so by also taking pushouts by arrows in Im one obtains all polygraphs with any possible marking on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Finally, it was shown in [23] that polygraphs are closed under retract in ∞-Cat, so they constitute all cofibrant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The pushout-product, or corner-product (sometimes also called Leibniz prod- uct) f ˆ → g and f ˆ ∼ g is defined as usual: if f: X → Y and g: A → B are two arrows in ∞-Catm, then f ˆ → g is the canonical arrow: X → B � X →A Y → A → Y → B and f ˆ ∼ g is the canonical arrow X ∼ B � X ∼ A Y ∼ A → Y ∼ B We refer to the appendix of [18] for the general theory of pushout products and their formal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='31 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If f and g are two cofibrations in ∞-Catm then f ˆ → g and f ˆ ∼ g are both cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' By the usual properties of the corner-product, it is enough to check this when f and g are generating cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If f and g are both in Ia, then f → g has no marked arrows in either its domain or codomain and coincides with the corner-product f ˆ⊗ g in ∞-Cat, which has been shown to be a cofibration in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' f ∼ g is the same except that some arrows are marked, but we can always add these marking by taking additional pushouts by arrows in Im, so it is again a cofibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 13 The forgetful functor ∞-Catm → ∞-Cat is monoidal for both tensor prod- uct and preserves colimits, so it preserves the corner-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In particular, if either f or g is in Im then it is sent to isomorphisms by this forgetful functor and hence f ˆ → g and f ˆ ∼ g induces isomorphisms between their underlying ∞- categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Now, if f: (X, N) → (X, M) is a morphism in ∞-Catm that induces an isomorphism on underlying ∞-categories, then it is a pushout of arrows in Im: one simply needs to take such pushout to make all arrows in M marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='32 Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We define I: = D♯ 1 = (D1, {e1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It is the ∞-category with two objects, e− 0 and e+ 0 and a marked arrow e1: e− 0 → e+ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We denote by j− and j+ the two maps D0 → I corresponding respectively to the two objects e− 0 and e+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This gives a diagram: D0 � D0 \u058c I → D0 Which will play the role of the interval object for our semi-model structure on ∞-Catm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will take as a set of “generating anodyne cofibrations” (also called a “pseudo-generating set of trivial cofibrations”) the set of maps of the form j+ ˆ ∼ i where i is a generating cofibration, more precisely: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='33 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that an arrow in ∞-Catm is a naive fibration if it has the right lifting property against all arrows of the form j+ ˆ ∼ i, where j+: D0 → I is as in Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='32, and i is one of the generating cofibrations as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that an arrow in ∞-Catm is an anodyne cofibrations if it has the right lifting property against all naive fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that a cofibration in ∞-Catm is acyclic if it has the lifting property against all naive fibrations between (naively) fibrant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that a map in ∞-Catm is a fibration if it has the right lifting property against all acyclic cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' As before, it immediately follows from the small object argument that every arrow factors as an anodyne cofibration followed by a naive fibration, and all anodyne cofibration are retracts of transfinite compositions of pushouts of the “generating anodyne cofibrations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='34 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It immediately follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='31 that, as j+ is a cofibration, all maps of the form j+ ˆ ∼ i are cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In particular, all trivial fibrations are also naive fibrations and all anodyne cofibrations are cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='35 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Acyclic cofibrations and fibrations form a cofibrantly gen- erated weak factorization system on ∞-Catm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An object is “naively fibrant” if and only if it is fibrant and more generally an arrow between fibrant objects is a fibration if and only if it is a naive fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is a direct application of the results of Section 4 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Starting from the premodel structure on ∞-Catm whose weak factorization systems are (cofibrations, trivial fibrations) and (anodyne cofibrations, naive fibrations), we obtain the one with (cofibrations, trivial fibrations) and (acyclic cofibrations, fibrations) as its “left saturation” L(∞-Catm) in the sense of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' All the claim in the proposition follows from this Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='36 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that replacing ˆ ∼ by ˆ → in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='33 would not change the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Indeed, if X = Y ♯ is an m-marked ∞-category whose arrows of dimension > 0 are all marked then for any m-marked ∞-category Z one has X ∼ Z = X → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' As this applies to both the domain and the co-domain of j+ it follows that j+ ˆ ∼ i = j+ ˆ → i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Also, the reader should not be worried about the use of j+ in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='33 rather than j− or both j− and j+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' While putting j− or both j− and j+ instead of j+ would change the definition of naive fibrations and anodyne cofibrations, this does not affect the definition of (naive) fibrations between fibrant objects, hence the acyclic cofibrations and fibrations would not be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Indeed, once the existence of a (monoidal) model structure is established, it follows that j− is acyclic by 2-out-of-3, and hence all the maps j− ˆ ∼ i = j− ˆ → i are also acyclic cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='37 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If f is an anodyne (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' acyclic) cofibration and g is a cofibra- tion then f ˆ ∼ g and f ˆ → g are anodyne (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' acyclic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' To get the result for “anodyne cofibrations” it is enough to prove it for the generating anodyne cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let i be one of the generating cofibrations and f = j+ ˆ ∼ i′ be one of the generating anodyne cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We have f ˆ ∼ i = j+ ˆ ∼ (i ˆ ∼ i′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' As i′ ˆ ∼ i is a pushout of generating cofibrations i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' , ik by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='31 it follows that j+ ˆ ∼ (i ˆ ∼ i′) is a pushout of the j+ ˆ ∼ ik and hence is an anodyne cofibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The result for acyclic cofibrations follows from formal properties of the pushout product: it follows that if i is a cofibration and p is a naive fibra- tion then the (right) pullback exponential ⟨p/i⟩ is a naive fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If p is a (naive) fibration between fibrant objects then ⟨p/i⟩ is a naive fibration between fibrant objects hence a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It follows that if i a acyclic cofibration and j is a cofibration then i ˆ ∼ j is an acyclic cofibration as it is a cofibration by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='27 and if p is a fibration between fibrant objects then i ˆ ∼ j has the right lifting property against p because j has the left lifting property against ⟨p/i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The case of → works exactly the same considering the first half of Re- mark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='38 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The category ∞-Catm of m-marked ∞-category admits a left semi-model structure, called the inductive model structure, in which the cofi- brations and trivial fibrations are as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='27 and the fibrations are as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This immediately follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='12 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Because of Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='31 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='37, tensoring by the interval object I of Construc- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='32 is a “strong Quillen functor” in the sense of section 6 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that to apply Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='12 one needs to observe that ∞-Catm, with the (cofibra- tions, trivial fibrations) and (acyclic cofibrations, fibrations) weak factorization systems, is both “right saturated” and “left saturated” that is, that a fibration that has the right lifting property against all cofibrations between cofibrant ob- jects is a trivial fibration and that a cofibration that has the left lifting property against all fibrations between fibrant objects is a trivial cofibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The first ones hold because the generating cofibrations are cofibrations between cofibrant objects and the second because that is how we defined acyclic fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='39 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='38 above also shows that ∞-Catm also admits a right semi-model category structure whose fibrations and trivial cofibrations are the fibrations and acyclic cofibrations of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='33 and whose cofibrations are as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This however does not clearly make ∞-Catm into a Quillen model struc- ture but rather into a “two-sided model category” as in Section 5 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We refer to Section 5 of [15] for what this means more precisely, but in short, the problem is that the left and right semi-model categories have different classes of weak equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The two classes of equivalence however coincide for arrows that are between fibrant or cofibrant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Another way to talk about this difference is that left and the right semi-model categories are Quillen equivalent and have the same homotopy category, but define different functors ∞-Catm → Ho(∞-Catm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The two functors agree on objects that are either fibrant or cofibrant but differ on general objects: one sends an object X to its cofibrant replacement while the other sends it to a fibrant replacement, and we do not know if these are always homotopy equivalent when X is neither fibrant nor cofibrant itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='40 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We do not know if ∞-Catm is actually a Quillen model cate- gory or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In the unmarked case, this follows from the fact that all objects are fibrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' But that is no longer the case in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In terms of the “two-sided model structure” mentioned in the previous remark, the question is whether ∞-Catm satisfies one of the equivalent conditions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We conclude this section with the following lemma that will be useful later: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='41 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The map i+ n : D♭ n → (Dn+1, {en+1}) where en+1 is the unique non-identity arrow of Dn+1, is an anodyne cofibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will show it is a retract of the map j+ ˆ ∼ in where in is the map ∂Dn → Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In order to achieve this, we will compute j+ ˆ ∼ in more explicitly using the description of D1 ⊗ Dn given in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 of [4] (see proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4): As a polygraph, the generating arrows of D1 ⊗ Dn are the: a− 0 ⊗ eǫ k a+ 0 ⊗ eǫ k a ⊗ eǫ k where the arrows of D1 have been denoted “a” instead of “e” to distinguish them, and ǫ is either + or −, k ⩽ n and e+ n = e− n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Their source and target are given as follows: π−(a− 0 ⊗ eǫ k) = a− 0 ⊗ e− k−1 π+(a− 0 ⊗ eǫ k) = a− 0 ⊗ e+ k−1 π−(a+ 0 ⊗ eǫ k) = a+ 0 ⊗ e− k−1 π+(a+ 0 ⊗ eǫ k) = a+ 0 ⊗ e+ k−1 π−(a ⊗ eǫ k) = (a− 0 ⊗ eǫ k)#0(a ⊗ e+ 0 )#1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' #k−1(a ⊗ e+ k−1) π+(a ⊗ eǫ k) = (a ⊗ e− k−1)#k−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' #1(a ⊗ e− 0 )#0(a+ 0 ⊗ eǫ k) 16 We did not put parenthesis in the expression above, to keep them shorter, the default convention is to do the composition #i in order of increasing values of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The last two equations are given by proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 of [4], though note that this reference is using a different convention than ours regarding the composition order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that the object we are interested in is I ∼ D♭ n which is the same polygraph endowed with the special marking where all the arrows a ⊗ eǫ k are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We then realize (Dn+1, {en+1}) as a retract of I ∼ D♭ n+1 as follows: We call i: (Dn+1, {en+1}) → I ∼ D♭ n the unique morphism sending en+1 to a ⊗ en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is well defined because a ⊗ en is a marked arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Next, we define a map p: I ∼ D♭ n → (Dn+1, {en+1}) by: p(aǫ 0 ⊗ eµ k) = eµ k if k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' p(aǫ 0 ⊗ en) = eǫ n p(a ⊗ eǫ k) = Ieǫ k if k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' p(a ⊗ en) = en+1 In order to check that this is well defined, we first need to check that this definition is compatible with the source and target given above, which follow from an immediate calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Then we need to show that this is compatible with the marking, which is the case as both Ieǫ k and en+1 are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Finally, the composite p ◦ i send the arrow en+1 to p(a ⊗ en) = en+1 and hence is the identity of Dn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' To conclude the proof, we just have to observe that the maps f and i defined above send the domain of i+ n and of j+ ˆ ∼ in to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The domain of j+ ˆ ∼ in is the sub-polygraph of I ∼ D♭ n which contains all the generators except a− 0 ⊗ en and a ⊗ en, while the domain of i+ n contains all generators of Dn+1 except en+1 and e− n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In order to check that the map i is compatible with these sub-polygraphs, it is enough to check that i(e+ n ) is in the domain of j+ ˆ ∼ in, to see this, we compute: i(e+ n ) = π+i(en+1) = π+(a ⊗ en) = (a ⊗ e− n−1)#n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' #1(a ⊗ e− 0 )#0(a+ 0 ⊗ en) and we observe that this expression involves neither a− 0 ⊗ en nor a ⊗ en, hence it does belong to the domain of j+ ˆ ∼ in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In order to check that the map p is compatible with these sub-polygraphs, we need to check the image by p of all the generators of I ∼ D♭ n except a− 0 ⊗ en and a⊗en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' These are given by the formulas p(aǫ 0⊗eµ k) = eµ k if k < n, p(a+ 0 ⊗en) = e+ n and p(a ⊗ eǫ k) = Ieǫ k, which all indeed belong to the image of i+ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3 Equations and saturations in an m-marked ∞- category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The general goal of this section is to arrive at a better description of the fibrant objects and fibrations between fibrant objects of the model structure of Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is achieved using the notion of “equations” in an ∞-categories introduced by the second named author in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will recall the basic theory of equations, in a slightly different language and introduce an analog of equations to deal with the markings, which we call saturations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 Definitions of equations and saturations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A left equation is a special m-marked polygraph (P, M) with two arrows x, y ∈ P such that: (1) y is the unique arrow of dimension n + 1 and P contains no arrows of dimension > n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (2) y is a marked arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (3) if n ≤ m, x is an unmarked arrow of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (4) The source of y admits a decomposition: π− n y = ln#n−1(ln−1#n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='#1(l1#0x#0r1)#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='#n−2rn−1)#n−1rn where for each i, li and ri are marked i-arrow in P, with ln and rn not containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In particular, x appears only once in π− n y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (5) x does not appear in the target of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Right equations are defined in the exact same way except the source and target of y are exchanged in the last two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that (P, M) is an equation to mean that it is either a left or right equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If P, with its arrows x and y as in the definition, is an equation one denotes by ΛP the sub-polygraphs of P that contains all arrow except x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that specifying the arrows x, y ∈ P is exactly the same as specifying the subpolygraphs ΛP ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For this reason, we will often also call “equation” the map ΛP → P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that an equation ΛP → P has solutions in C ∈ ∞-Catm if C has the right lifting property against ΛP → P and we say that a morphism f: C → D lifts solutions of the equation if it has the right lifting property against the map ΛP → P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The name “equation” comes from the idea that we are looking for an element x such that a certain composite of x with other arrows is isomorphic to another given arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' From this point of view, a map ΛP → X corresponds to such an equation in X, and an extension P → X corresponds to a solution of the equation, or rather the image of x is the solution and y represents the isomorphism witnessing that x is a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A left saturation is a special marked polygraph (P, M) with arrows x and y satisfying the conditions of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 except that x is a marked arrow and the target of y is a marked arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Right saturations are defined in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If P is a saturation, one denotes ΩP the special m-marked polygraph (P, M − {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='5 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If P is an equation, we define the m-marked ∞-category Uni(P) which is the colimit of the following diagram: ∂Dn D♯ n P � ΛP P Uni(P) x∐x′ z ⌟ 18 A map Uni(P) → X corresponds to a map ΛP → X, which is an equation in X, together with two solutions P → X, given by pairs (x, y) and (x′, y′), and a marked arrow z: x → x′ which express that the two solutions are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='6 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let C be an m-marked ∞-category C and P a left equation (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' right equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The equation P has solutions in C if for all morphisms ΛP → C, there exists a lifting (x, y): P → C such that x is sent on a marked arrow whenever the target of y is (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' the source of y is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Solutions to an equation P are C are weakly unique if C has the right lifting property against P � ΛP P → Uni(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The equation P has unique solutions in C if the equation P has solutions in C and they are weakly unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It will be useful to have a “coherent” version of Uni(P), noted Unicoh(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If P is a left equation, P � ΛP P → Unicoh(P) is obtained as the following sequence of pushout: ∂Dn Dn P � ΛP P Unicoh(P) ∂Dn+1 Dn+1 x∐x′ z ⌟ s[x/z]#ny′∐y ⌟ where s is the source of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Conversely, if P is a right equation, P � ΛP P → Unicoh(P) is obtained as the following sequence of pushout: ∂Dn Dn P � ΛP P Unicoh(P) ∂Dn+1 Dn+1 x∐x′ z ⌟ y#nt[x/z]∐y′ ⌟ where t is the source of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Remarks that in both cases, P � ΛP P → Unicoh(P) is an equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' By definition, if C is an m-marked ∞-category such that Unicoh(P) has a solution in C, then C as the right lifting property against P � ΛP P → Uni(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='7 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The morphism j+ ˆ ∼ in: = I ∼ ∂Dn � {1} ∼ Dn → I ∼ Dn is a left equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Indeed, let y be the top dimensional generator of I ∼ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If we denote by x the top dimensional arrow of {0} ∼ Dn, and for 0 < k ≤ n, by ak the image of the top dimensional k-generator of I ∼ Dk−1 by the morphism I ∼ δ− k−1: I ∼ Dk−1 → I ∼ Dn, 19 Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' of [4] allows to give an explicit description of I ∼ Dn, which we recalled in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Using this description, we see that if we name y = a ⊗ en and x = a− 0 ⊗ en the two arrows of I ∼ Dn that are not in the image of j+ ˆ ∼ in, then we have a decomposition of the source of y of the form: (((x#0a0)#1a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=')#n−1an and all the ak are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We denote it eq n : ΛEq n → Eq n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='8 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Similarly, the morphism j+ ˆ ∼ sn: I ∼ Dn � {1} ∼ (Dn, {en}) → I ∼ (Dn, {en}) where sn is the “identity” map Dn → (Dn, {en}) is a left saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' which we denote sat n : ΩSat n → Sat n 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='9 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We define some left equations which play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In each case, k and n are integers with k ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' eq k,n : ΛEq k,n → Eq k,n , whose target is generated by x and b of di- mension n, a a marked arrow of dimension k and y: (a#k−1x) ⇒ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' eq k,n : ΛEq k,n → Eq k,n , whose target is generated by x and b of di- mension n, a a marked arrow of dimension k and y: (x#k−1a) ⇒ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In all equations above, the domain of the arrow is obtained by removing x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Also, in each case, we have not listed all the constraints of the source and target that are necessary to make sense of the definition of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For example, in Eq k,n , we have the relation π+ k−1(a) = π− k−1(x) for the composition a#k−1x to exists, and the relations πǫ n−1(b) = πǫ n−1(a#k−1x), as b needs to be parallel to a#k−1x for y to exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Characterization of fibrant objects In this section, we will give a simple characterization of the fibrant objects of the model structure introduced in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will temporarily call the objects satisfying this characterization “prefibrant” (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='12) and then show in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='19 that these are exactly the fibrant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='10 Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Suppose given an equation P and a lifting problem of the form: ΛP C P D p Given a a generator of P, we will denote its image in D also by a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If a ∈ ΛP, we denote by a its image in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' So in general p(a) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If the dotted diagonal lift exists, or in the process of constructing such a lift, the image of x, y ∈ P in C are also denoted x and y, and we hence also have p(x) = x and p(y) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='11 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let a be a (n + 1)-arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An inverse for a is an arrow a−1 such that there exist two marked arrows: ǫ: a#na−1 → I ν: a−1#na → I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An arrow is invertible if it has an inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='12 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An m-marked ∞-category C is prefibrant if (1) marked arrows are invertible and their inverses are marked, (2) whenever a and c: a → b are marked, so is b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This directly implies that if b and c: a → b are marked, so is b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This notion is purely temporary: we will show in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='19 that an object is fibrant for the model structure of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='38 if and only if it is prefibrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='13 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If C is prefibrant, then equations Eq k,n and Eq k,n have weakly unique solutions in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We show the result by a decreasing induction on k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The initialization corresponds to k = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In this case, the data of a morphism ΛEq n,n → C corresponds to two n-arrows a and b sharing the same source and such that a is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let ν: a−1#na → I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If we define x: = a−1#nb and y: ψ#nb: a#nx → b, the couple (x, y) is a solution of Eq n,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If b is marked so is x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We now show the weak unicity of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let (¯x, ¯y) be another solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We then have a marked arrow: z: ¯x ν−1 −−→ a−1#na#n¯x ¯y−→ a−1 ∗ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The assertion for Eq n,n is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Suppose now the result is true for all k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We start by showing that solutions of Eq k,n and Eq k,n are weakly unique in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The data of a morphism ΛEq k,n → C corresponds to an n-arrow x: s → t, a k-invertible arrow a, and an arrow b: a#k−1s → a#kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let (x, y: a#k−1x → b) be a solution of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let ν: a−1#na → I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The arrow x is then also a solution of Eq k+1,n: (ν#0s)#kx = (a−1#k−1a#k−1x)#k(ν#k−1t) (a−1#k−1b)#k(ν#k−1t) (a−1#k−1y)#k(ν#k−1t) and so is weakly unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The unicity of solution of Eq k,n is proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We show now that Eq k,n and Eq k,n have solutions in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let (x, y) be a solution of the equation (ν#0s)#kx (a−1#k−1b)#k(ν#k−1t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' y Moreover, we can find such x marked whenever b is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We then have (ν#0s)#kx = (a−1#k−1a#k−1x)#k(ν#k−1t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 21 By weakly unicity of solution of Eq k+1,n, we then have a marked arrow z: a−1#k−1a#k−1x → a−1#k−1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' But a#k−1x and b are solutions of an equation Eq k,n , and so there exist a marked arrow ˜y: a#k−1x → b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If b is marked, the arrow x that we produce is also marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The existence of solution of Eq k,n is proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='14 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If equations Eq k,n and Eq k,n have solutions in C, then all equations have solutions in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let P be a left equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' There is a decomposition of the source of y of the shape π− n y = ln#n−1(ln−1#n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='#1(l1#0x#0r1)#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='#n−2rn−1)#n−1rn where for each i, li and ri are marked i-arrow in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We can then use the existence of solutions to Eq k,n and Eq k,n to get two sequences of arrows (xk)0 0.3) while the most stable +(휎 < 0.1) are 418849 and 2018 EB. The presence of an as- +teroid in the co-orbital resonance appears, therefore, to be a +poor indicator of dynamical stability. Instead, the individ- +ual dispersions reported in Table 4 are correlated with the +orbital actions, so that low 푒 and high 퐼 orbits are the most +stable. +This suggests that the principal cause of instability is +orbit-changing planetary encounters. Indeed, a high orbit in- +clination helps to avoid frequent close encounters with plan- +ets and moderates their orbit-changing effects even when +they do occur (Michel and Thomas, 1996). Furthermore, +an asteroid in an orbit with 푒 ≳ 0.25 may approach Venus +as well as the Earth. On the other hand, resonance-locking +episodes and passing orbit states typically last ∼104 yr (Fig- +ure 11, see also Morais and Morbidelli (2002)) and much +shorter than a Myr. +In this narrative, 138175 diffuses the fastest due to its +high eccentricity allowing close encounters with Venus that +disrupt the co-orbital resonance. In the same way, the mod- +erate orbital diffusion displayed for 2016 CA138 and 2017 +SL16 is caused by deterministic changes while in the 1:1 +resonance with Earth and intermittent forays of the orbital +eccentricity above the Venus-crossing value. Finally, the +low-푒, high-퐼 orbits of 418849 and 2018 EB mitigate against +close approaches to both Earth and Venus. +We return now to the question posed in the Introduction, +namely whether orbital stability and the rotational state of +NEAs at 1 au are related. At first glance, this does not appear +to be the case since, for example, the set of six asteroids near +the spin rate barrier (Figure 11) is a mixture of stable and +unstable cases (Table 4). +Alternatively, we can choose to consider only those as- +teroids at the two extremes of the stability spectrum, ie those +with the highest and lowest values of 휎. If a relation of the +type we are searching for existed, we would expect the rota- +tional properties of those two groups to differ the most. +Here we find see that the spin rates of 418849, 468910 +and 2018 EB are near-critical while that of 138175 is sub- +critical. One might regard this as evidence that the vari- +able dynamical environment of 138175 interferes with the +spinning-up action of the YORP torque, preventing it from +reaching a near-critical spin state. In a sample of only four, +however, this evidence is far from conclusive and, while such +a relationship might still exist, extending our study to a much +larger sample of asteroids appears necessary to eke it out of +the data. +6. Conclusions +We determine the rotational periods and axis orienta- +tions of four Earth co-orbital asteroids. +The smallest object - 2017 SL16 with a diameter of a +few tens of meters - has a rotational period P=0.3188 hrs +which suggests that it is monolithic. Even though we can- +not find a definitive pole solution, we can say that it is likely +not near the ecliptic and the best solution from the five mod- +els with similar 휒2 values has coordinates 휆1=190.44◦ and +훽1=34.32◦ +The other extreme case is the asteroid (138175) 2000 +EE104, which we find to be a slow rotator with a spin rate of +P=13.9476 hrs. The pole position is also not uniquely deter- +mined but most probably it lies above the ecliptic with the +formal best-fit pole at 휆1=233.86◦ and 훽1=68.71◦. +G. Borisov et al.: Preprint submitted to Elsevier +Page 10 of 11 + +Earth co-orbitals properties +The object with the best quality of observations and re- +spectively the best results is (418849) 2008 WM64. It has +only one clear definitive solution for the rotational period: +P=2.4077 hrs. Also, the pole solution lies on a single spot of +the 휒2 search map at coordinates 휆1=217.34◦ and 훽1=-39.17◦. +For the second smallest object in our sample - 2016 CA138 +with an estimated diameter of 75 m - we determine the spin +rate to be 5.3137 hrs. The 휒2 map shows two vertical stripes +with low 휒2 values at longitudes 180◦ apart - around 80◦ and +260◦. +We compared the rotational properties and sizes of our +asteroid sample to NEA entries in the LCDB database (Warner, +B. D. and Harris, A. W. and Pravec, P., 2021). Our overall +conclusion is that the size vs rotational frequency distribu- +tion of the co-orbital asteroids appears not to differ from the +one of the NEAs. +We made numerical simulations in order to investigate +if there is a relation between the orbit stability and the ro- +tational state of Earth co-orbitals. Our results show that the +co-orbital resonance is not affecting the orbit stability, but +the orbit itself is responsible for that and mainly its eccen- +tricity (푒) and inclination (퐼). Orbits with low 푒 and high 퐼 +are the most stable because firstly highly-inclined orbits are +relatively stable against close encounters with planets and +secondly orbits with high 푒 may approach Venus as well as +the Earth. +We cannot make a definitive conclusion if the orbit sta- +bility and rotational state of the asteroids are related, so we +need further investigations and observations to increase our +sample in order to obtain more statistically significant re- +sults. +7. Acknowledgements +This work was supported via grant ST/R000573/1 from +the UK Science and Technology Facilities Council. The au- +thors gratefully acknowledge observing grant support from +the Institute of Astronomy and National Astronomical Ob- +servatory, Bulgarian Academy of Sciences. Astronomical +research at the Armagh Observatory & Planetarium is grant- +aided by the Northern Ireland Department for Communities +(DfC). The authors also acknowledge DfC for the FoReRo2 +instrument development contribution from the Armagh Ob- +servatory & Planetarium toward the new CCD camera Andor +iKon-L used in this study. +References +Borisov, G., Christou, A.A., Bagnulo, S., Cellino, A., 2021. Lightcurve +and spin rates of Earth co-orbital asteroids. Minor Planet Bulletin 48, +268–271. +Chambers, J.E., 1999. A hybrid symplectic integrator that permits close +encounters between massive bodies. MNRAS 304, 793–799. doi:10. +1046/j.1365-8711.1999.02379.x. +Christou, A.A., 2000. A numerical survey of transient co-orbitals of the +terrestrial planets. Icarus 144, 1–20. +Di Ruzza, S., Pousse, A., Alessi, E.M., 2023. On the co-orbital asteroids in +the solar system: medium-term timescale analysis of the quasi-coplanar +objects. Icarus 390, id. 115330. +Duddy, S.R., Lowry, S.C., Wolters, S.D., Christou, A., Weissman, P., +Green, S.F., Rozitis, B., 2012. +Physical and dynamical characteri- +sation of the unbound asteroid pair 7343-154634. +A&A 539, A36. +doi:10.1051/0004-6361/201118302. +Durech, J., Kaasalainen, M., Warner, B.D., Fauerbach, M., Marks, S.A., +Fauvaud, S., Fauvaud, M., Vugnon, J.M., Pilcher, F., Bernasconi, +L., Behrend, R., 2009. +Asteroid models from combined sparse and +dense photometric data. A&A 493, 291–297. doi:10.1051/0004-6361: +200810393. +Durech, J., Sidorin, V., Kaasalainen, M., 2010. DAMIT: a database of as- +teroid models. A&A 513, A46. doi:10.1051/0004-6361/200912693. +Fenucci, M., Novaković, B., 2020. The role of the Yarkovsky Effect in the +long-term dynamics of asteroid (469219) Kamo’oalewa. Astron. J. 162, +id. 227. doi:10.3847/1538-3881/ac2902. +Giorgini, J.D., Yeomans, D.K., Chamberlin, A.B., Chodas, P.W., Jacobson, +R.A., Keesey, M.S., Lieske, J.H., Ostro, S.J., Standish, E.M., Wimberly, +R.N., 1996. JPL’s on-Line Solar System data service, in: AAS/Division +for planetary sciences meeting abstracts #28, p. 25.04. +Harris, A.W., Pravec, P., 2005. Rotational properties of asteroids, comets +and TNOs, in: Symposium S229: Asteroids, Comets, Meteors, pp. 439– +447. +Jacobson, S.A., 2014. Small asteroid system evolution, in: Complex Plan- +etary Systems, pp. 108–117. +Jacobson, S.A., Scheeres, D.J., 2011. Dynamics of rotationally fissioned +asteroids: Source of observed small asteroid systems. Icarus 214, 161– +178. doi:10.1016/j.icarus.2011.04.009, arXiv:1404.0801. +Jewitt, D., 2020. 138175 (2000 EE104) and the source of interplanetary +field enhancements. The Planetary Science Journal 1, 33. doi:10.3847/ +PSJ/aba68f, arXiv:2007.07192. +Kaasalainen, M., Torppa, J., Muinonen, K., 2001. +Optimization meth- +ods for asteroid lightcurve inversion. II. The complete inverse problem. +Icarus 153, 37–51. doi:10.1006/icar.2001.6674. +Kaplan, M., Cengiz, S., 2020. Horseshoe co-orbitals of Earth: current pop- +ulation and new candidates. MNRAS 496, 4420–4432. doi:10.1093/ +mnras/staa1873, arXiv:2006.14451. +Lai, H.R., Russell, C.T., Wei, H.Y., Connors, M., Delzanno, G.L., +2017. +Possible potentially threatening co-orbiting material of aster- +oid 2000EE104 identified through interplanetary magnetic field distur- +bances. Meteoritics and Planetary Science 52, 1125–1132. doi:10.1111/ +maps.12854. +Michel, P., Thomas, F., 1996. The Kozai resonance for near-Earth asteroids +with semimajor axes smaller than 2AU. A&A 307, 310. +Morais, M.H.M., Morbidelli, A., 2002. The population of near-Earth aster- +oids in coorbital motion with the Earth. Icarus 160, 1–9. doi:10.1006/ +icar.2002.6937. +Namouni, F., 1999. Secular interactions of coorbiting objects. Icarus 137, +293–314. doi:10.1006/icar.1998.6032. +Pravec, P., Harris, A.W., Kušnir ak, P., Galád, A., Hornoch, K., 2012. Abso- +lute magnitudes of asteroids and a revision of asteroid albedo estimates +from WISE thermal observations. Icarus 221, 365–387. +Pravec, P., Vokrouhlický, D., Polishook, D., Scheeres, D.J., Harris, A.W., +Galád, A., Vaduvescu, O., Pozo, F., Barr, A., Longa, P., Vachier, F., +Colas, F., Pray, D.P., Pollock, J., Reichart, D., Ivarsen, K., Haislip, J., +Lacluyze, A., Kušnirák, P., Henych, T., Marchis, F., Macomber, B., Ja- +cobson, S.A., Krugly, Y.N., Sergeev, A.V., Leroy, A., 2010. Forma- +tion of asteroid pairs by rotational fission. +Nature 466, 1085–1088. +doi:10.1038/nature09315, arXiv:1009.2770. +Qi, Y., Qiao, D., 2022. Stability analysis of Earth co-orbital objects. As- +tron. J. 163, id. 211. +Rowe, B., 2018. Lightcurve analysis of 6 asteroids from RMS observatory. +Minor Planet Bulletin 45, 292–294. +Rubincam, D.P., 2000. Radiative spin-up and spin-down of small asteroids. +Icarus 148, 2–11. +Scheeres, D.J., Marzari, F., Rossi, A., 2004. Evolution of NEO rotation +rates due to close encounters with Earth and Venus. Icarus 170, 312– +G. Borisov et al.: Preprint submitted to Elsevier +Page 11 of 11 + +Earth co-orbitals properties +Asteroid #138175 +-0.09 +-0.06 +-0.03 +-0.00 +0.03 +0.06 +0.09 +0.12 +Rel. Semimajor Axis +0.25 +0.28 +0.31 +0.34 +0.37 +0.40 +Eccentricity +-10000 -8000 -6000 -4000 -2000 +0 +2000 +4000 +6000 +8000 10000 +time (yr) +0 +2 +4 +6 +8 +Inclination (deg) +Asteroid #138175 +-0.4 +-0.2 +0.0 +0.2 +0.4 +Rel. Semimajor Axis +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Eccentricity +-1×106 -8×105 -6×105 -4×105 -2×105 +0 +2×105 4×105 6×105 8×105 1×106 +time (yr) +0 +4 +8 +12 +16 +20 +24 +Inclination (deg) +Asteroid #418849 +-0.004 +-0.002 +0.000 +0.002 +0.004 +0.006 +0.008 +Rel. Semimajor Axis +0.00 +0.04 +0.08 +0.12 +0.16 +0.20 +Eccentricity +-10000 -8000 -6000 -4000 -2000 +0 +2000 +4000 +6000 +8000 10000 +time (yr) +31 +32 +33 +34 +35 +Inclination (deg) +Asteroid #418849 +-0.12 +-0.09 +-0.06 +-0.03 +0.00 +0.03 +0.06 +0.09 +0.12 +Rel. Semimajor Axis +0.0 +0.1 +0.2 +0.3 +0.4 +Eccentricity +-1×106 -8×105 -6×105 -4×105 -2×105 +0 +2×105 4×105 6×105 8×105 1×106 +time (yr) +25 +27 +29 +31 +33 +35 +37 +Inclination (deg) +Asteroid 2016 CA138 +-0.008 +-0.006 +-0.004 +-0.002 +0.000 +0.002 +0.004 +0.006 +0.008 +Rel. Semimajor Axis +0.00 +0.03 +0.06 +0.09 +0.12 +0.15 +0.18 +Eccentricity +-10000 -8000 -6000 -4000 -2000 +0 +2000 +4000 +6000 +8000 10000 +time (yr) +26.0 +26.5 +27.0 +27.5 +28.0 +28.5 +29.0 +Inclination (deg) +Asteroid 2016 CA138 +-0.2 +-0.1 +0.0 +0.1 +0.2 +Rel. Semimajor Axis +0.0 +0.1 +0.2 +0.3 +0.4 +Eccentricity +-1×106 -8×105 -6×105 -4×105 -2×105 +0 +2×105 4×105 6×105 8×105 1×106 +time (yr) +13 +16 +19 +22 +25 +28 +31 +34 +Inclination (deg) +Asteroid 2017 SL16 +-0.008 +-0.006 +-0.004 +-0.002 +0.000 +0.002 +0.004 +0.006 +0.008 +Rel. Semimajor Axis +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Eccentricity +-10000 -8000 -6000 -4000 -2000 +0 +2000 +4000 +6000 +8000 10000 +time (yr) +0 +3 +6 +9 +12 +15 +Inclination (deg) +Asteroid 2017 SL16 +-0.3 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +Rel. Semimajor Axis +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Eccentricity +-1×106 -8×105 -6×105 -4×105 -2×105 +0 +2×105 4×105 6×105 8×105 1×106 +time (yr) +0 +3 +6 +9 +12 +15 +18 +Inclination (deg) +Figure 11: Relative semimajor axis (푎 − 푎Earth)∕푎Earth, eccentricity 푒 and inclination 퐼 relative to the ecliptic plane for the nominal +orbit and 20 clones of each asteroid over 104 yr (left) and 106 yr (right) from 푡 = 0. Black dashed and solid lines represent the +mean and standard deviation for each clone set. The red plus symbol indicates the starting orbit. +G. Borisov et al.: Preprint submitted to Elsevier +Page 12 of 11 + +Earth co-orbitals properties +Asteroid 2016 CA138 +-1000-800 -600 -400 -200 +0 +200 400 600 800 1000 +time (yr) +-180 +-120 +-60 +0 +60 +120 +180 +Resonant angle (degrees) +Asteroid 2017 SL16 +-1000-800 -600 -400 -200 +0 +200 400 600 800 1000 +time (yr) +-180 +-120 +-60 +0 +60 +120 +180 +Resonant angle (degrees) +Figure 12: Evolution of the resonant angle Δ휆 = 휆 − 휆Earth +for the clones of Earth co-orbitals 2016 CA138 (top) and 2017 +SL16 (bottom) and for 103 yr backwards and forwards from +the present. The dashed horizontal line and red plus symbol +indicate the Δ휆 = 0◦ datum and the starting orbit respectively. +323. +Tricarico, P., 2017. The near-Earth asteroid population from two decades of +observations. Icarus 284, 416–423. doi:10.1016/j.icarus.2016.12.008, +arXiv:1604.06328. +Walsh, K.A., Richardson, D.C., 2006. A binary near-Earth asteroid forma- +tion: Rubble pile model of tidal disruptions. Icarus 180, 201–216. +Walsh, K.A., Richardson, D.C., Michel, P., 2008. Rotational breakup as the +origin of small binary asteroids. Nature 454, 188–191. +Warner, B.D., 2018. Near-Earth asteroid lightcurve analysis at CS3-Palmer +Divide station: 2017 October-December. Minor Planet Bulletin 45, 138– +147. +Warner, B. D. and Harris, A. W. and Pravec, P., 2021. Asteroid lightcurve +database (LCDB) bundle V4.0. NASA planetary data system. URL: +https://doi.org/10.26033/j3xc-3359. +Wiegert, P.A., Innanen, K.A., Mikkola, S., 1997. An asteroidal companion +to the Earth. Nature 387, 685–686. +G. Borisov et al.: Preprint submitted to Elsevier +Page 13 of 11 + diff --git a/FdE1T4oBgHgl3EQfqwVD/content/tmp_files/load_file.txt b/FdE1T4oBgHgl3EQfqwVD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6453b57e8d83b6bf0eb2be2ffb5db82c0f5d010 --- /dev/null +++ b/FdE1T4oBgHgl3EQfqwVD/content/tmp_files/load_file.txt @@ -0,0 +1,1607 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf,len=1606 +page_content='Physical and dynamical properties of selected Earth co-orbital asteroids⋆ Galin B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisova,b,∗,1, Apostolos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Christoua and Gordana Apostolovskac aArmagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, Northern Ireland, United Kingdom bInstitute of Astronomy with NAO, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussée Blvd, BG-1784, Sofia, Bulgaria cInstitute of Physics, Faculty of Natural Sciences and Mathematics, Ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Cyril and Methodius University in Skopje, Arhimedova 3, 1000 Skopje, FYR of Macedonia A R T I C L E I N F O Keywords: asteroids NEAs Earth co-orbitals photometry light curve numerical dynamical simulations A B S T R A C T We present our investigations of the physical and dynamical properties of selected Earth co-orbital asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The photometric optical light curves as well as rotation periods and pole solutions for a sample of four Earth co-orbital asteroids, namely (138175) 2000 EE104 (P=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='9476±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0051 hrs), (418849) 2008 WM64 (P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4077±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0001 hrs), 2016 CA138 (P=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3137±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0016 hrs) and 2017 SL16 (P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3188±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0053 hrs), are determined or improved and presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' For this investi- gation, we combine observations carried out at the Bulgarian National Astronomical Observatory - Rozhen using the FoReRo2 instrument attached to the 2mRCC telescope as well as sparse data from AstDys2 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Parallel to the rotational properties we did numerical dynamical simulations to investigate the orbital stability of those objects and to find out if there is a relation with their rotational properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Our results show that the orbit stability is affected by the orbit itself and mainly its eccen- tricity and inclination, which are responsible for the close encounters with other Solar system planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We cannot make a definitive conclusion about the relation between orbit stability and the rotational state of the asteroids, so we need further investigations and observations in order to prove or disprove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Introduction Asteroids with an average heliocentric distance of 1 au also called the Earth co-orbital asteroids - present a spe- cial challenge to Earth-based surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Because of the very slow net relative motion with respect to the Earth from one orbital revolution to the next, they usually remain far from our planet but close to the Sun’s projection to the sky for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' This effect leads to a much lower observational completeness for these types of objects than for other near- Earth asteroids (NEAs) (Tricarico, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' When a co-orbital object is discovered, it typically offers a few brief annually- recurring apparitions when it is bright enough to allow phys- ical characterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Afterwards, the accumulated Keple- rian drift due to the slight difference in orbital frequency be- tween Earth and the asteroid will place it out of reach of observational scrutiny for many decades hence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The dynamical behaviour of Earth coorbitals continues to be an area of active research (Di Ruzza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Qi and Qiao, 2022) ever since the discovery of the first member of this class, (3753) Cruithne (Wiegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 1997) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Co- orbital asteroids are generally thought to have more stable orbits against encounters with the planets, where “stable” is here taken to mean that the semimajor axis 푎, eccentricity 푒 and inclination 퐼 diffuse slowly over time or not at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' This is partly due to relatively long intervals between planetary conjunctions but also because, even for those asteroids that ⋆Partially based on data collected with the 2-m RCC telescope at the Bulgarian National Astronomical Observatory - Rozhen ∗Corresponding author Galin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='Borisov@Armagh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='uk (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov) ORCID(s): 0000-0002-4516-459X (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov) approach the planet, the resonant condition generally yields only shallow encounters where the resulting orbit change is both small and deterministic, leading to so-called transitions between coorbital modes (Namouni, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Christou, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The principal driver of rotational and physical changes in main belt asteroids smaller than ∼6 km is the non-gravitational YORP effect, causing significant changes to the spin state over a timescale that is size-dependent but typically <107 yr (Rubincam, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Jacobson, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In the terrestrial planet region, YORP competes with the torques and tides exerted across the asteroid during close planetary encounters (Scheeres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Walsh and Richardson, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Walsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' It stands therefore to reason that, if close encounters are im- portant to the physical and rotational evolution of NEAs, the properties of Earth co-orbitals might differ from those of other NEAs as the resonant condition offers some degree of protection from the closest encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In this paper, we report on rotational state of a sample of Earth co-orbital asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Of the four co-orbital NEAs in our sample, two were the subject of previous work (Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2021) while the remaining two are investigated here for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We generate rotational pole solutions for all four asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In parallel, we carry out orbit simulations of these four objects to establish which are actually locked in the 1:1 mean motion resonance with the Earth and quan- tify their long-term orbital stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Afterwards, we use this information to verify, in the first instance, whether the co- orbital state promotes long-term stability of the orbit and, secondly, to compare the ensemble rotational data and or- bital data between different populations: resonant as well as non-resonant asteroids at 1 au and NEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The paper is organised as follows: in the next section, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 1 of 11 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='03346v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='EP] 9 Jan 2023 aEarth co-orbitals properties Table 1 Observing circumstances and aspect data Number Designation yyyy mm dd Phase(◦)a LPAB b BPAB c Grpd (418849) 2008 WM64 2017 12 25 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='9 96 19 APO (138175) 2000 EE104 2018 11 09 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 100 8 APO — " — — " — 2019 01 01 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 108 14 APO — " — — " — 2020 01 02 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='9 104 14 APO 2017 SL16 2020 09 22 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 14 6 ATE 2016 CA138 2020 02 17&18 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 158 7 ATE aThe phase angle is given for the first date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' bThe approximate phase angle bisector longitude at mid-date range cThe approximate phase angle bisector latitude at mid-date range dThe asteroid family/group (APO-Apollo, ATE-Aten) Figure 1: Phase angle distribution of all the observations (dense and sparse) for each of our four asteroids (first row) and their corresponding phase angle bisector longitude (PABL) (second row) we present the photometric observations utilised in the ro- tational state modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Section 3 describes the data reduc- tion with emphasis on the different approaches used to es- timate the rotational state, while Section 4 presents our re- sults separately for the four asteroids and compares them to the NEA population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Section 5 is dedicated to the numeri- cal simulations to investigate the orbit dynamical properties and search for relationships to the rotational state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Finally, Section 6 outlines our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Observations Photometric observations of selected Earth co-orbital as- teroids were carried out from the Bulgarian National Astro- nomical Observatory - Rozhen, using the Two-channel Fo- cal Reducer Rozhen or “FoReRo2” instrument attached to the 2-m RCC telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The asteroid targets and their ob- servational circumstances are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In addition, we are considering available sparse data on the asteroids taken from the AstDys-2 database1, choosing to use only those measurements with a reported accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='01 magnitude or higher (see Table 2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' This helps us to extend the observational circumstances and es- pecially the phase angle to a much larger range (see Figure 1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Because sparse data is sporadic and with lower photometric accuracy than the dense data as they are mainly astrometry measurements and photometry is a secondary re- sult, we are using those data with a lower weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 in- stead of the value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 used for dense data (Durech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' With such a collection of dense and sparse data we can better determine the rotational period and the spin axis orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 1https://newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='spacedys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='com/astdys/ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier ' metadata={'source': 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+page_content='±180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='+10° ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='06+138175 2000 EE104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='Phase Angle Distribution: Minus: pre-opp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='Plus: post-opp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='90 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='270138175 2000 EE104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='PAB Longitude Distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2702016 CA138 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='PAB Longitude Distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2702017 SL16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='PAB Longitude Distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='90 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='I41g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2020 02 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='I41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='aMinor Planet Center (MPC) observatory codes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='bT08 – Asteroid Terrestrial-impact Last Alert System (ATLAS-MLO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' at Mauna Loa Observatory) cT05 – Asteroid Terrestrial-impact Last Alert System (ATLAS-HKO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' at Haleakala Observatory) dG96 – Mount Lemmon Survey e703 – Catalina Sky Survey, Tucson fI52 – Mount Lemmon Observatory (CHECK: Mount Lemmon Sur- vey) of the Steward Observatory gI41 – Zwicky Transient Facility (ZTF) and its predecessor Palomar Transient Factory (PTF) at Palomar Observatory 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Data reduction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Rotational period determination The data were reduced by applying standard bias and flat-field corrections followed by aperture photometry to pro- duce the light curve for each of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' To determine the rotational period of the asteroids, in- stead of resorting to a standard Fourier analysis or investi- gating the 휒2 of the fitted observational data by a Fourier function with different orders, we used a light curve inver- sion method to determine a simple 3D shape model of the object that reproduces the modelled light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Then we compare this light curve to the observational data to obtain the best period solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' For our purposes, we used the soft- ware provided by the Database of Asteroid Models from In- version Techniques (DAMIT2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The software was developed by Mikko Kaasalainen in Fortran and converted to C by Josef Durech (Durech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' For period determination we are using the sub-routine period_scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' To ensure that the global minimum of 휒2 in the period search is not missed, we scan through a fairly wide interval of possible periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Start- ing with six initial poles for each trial period and selecting the period that gives the lowest 휒2, if there is a clear mini- mum in 휒2 when plotted as a function of period we choose this solution as the best period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' If there is no clear minimum in the 휒2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' period plot or that many pole solutions are giving the same residual – it means that there is not enough data for a unique model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' According to Kaasalainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' (2001), the smallest sep- aration Δ푃 of the local minima in the trial period 푃 spectrum of the 휒2 of the light curve fit is roughly given by Δ푃 푃 ≈ 1 2 푃 푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' (1) where 푇 = max(|푡 − 푡0|) within the light curve set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' the timespan of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Kaasalainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' (2001) also explain that the period uncertainty is mostly governed by the epochs of the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' If the best local 휒2 minimum of the period spec- trum is clearly lower than the others, one can obtain an error estimate of, say, a hundredth part of the smallest minimum width Δ푃 since the edge of a local minimum ravine always lies much higher than its bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Thus the period determina- tion can be very accurate for data that cover many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' On the other hand, if the neighbouring minima are not clearly higher than the best one, the accuracy cannot be considered better than Δ푃 since the local error estimate cannot be ap- plied globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Spin axis orientation To determine a rotational pole solution for the asteroids, we ran the convexinv routine with different initial poles ran- domly distributed over the unit sphere and with 15 deg steps in both ecliptic longitude (휆) and latitude (훽) to produce 312 initial pole orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In such a way we are construct- ing a 휒2-map using all the solutions (top panel of each of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 5,3,7 & 9) which we are using to isolate the areas with low 휒2 and to find the best one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' If there is one pole solution that gives significantly lower 휒2 than all others, we adopt this pole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' For asteroids orbiting near the ecliptic plane, there are always two possible poles with the same 훽 and 휆±180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Objects with no previous period solution Here we are presenting results for objects in our sample with no previously published rotational information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 3 of 11 Earth co-orbitals properties 2017 SL16 2017 SL16 (H=25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8) is the smallest object we observed based on its absolute magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The light curve analysis shows that it is a very fast rotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The 휒2 variation with probe periods calculated by comparing the observations to the modelled light curves obtained from the simple 3D shape model is presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The period with the lowest 휒2 is P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3188 hrs (ΔP=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0053 hrs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The other 휒2 minima are multiples of the selected one, our decision to adopt this solution is based on the phased light curve having a shape with two minima and two maxima (middle panel of Fig- ure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Using the obtained rotational period we continue further by using it as an input value for a pole orientation search procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The result is shown at the top panel of Figure 3 as a 휒2-map which was constructed as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The corresponding model fitting to the obser- vational data and residuals are presented on the middle and bottom panels of the same figure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The first five pole solutions with 휒2 less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We cannot make a conclusive decision which one is correct, but they all have similar 휒2 values and give very similar rota- tion periods and those marked "3" and "3m" could be mirror solutions as explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Additional constraints on the 2https://astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='troja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='mff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='cuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='cz/projects/damit/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 Probe Period, hours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 χ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='34 Probe Period, hours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 χ2 Figure 2: 2017 SL16 - 휒2 vs probe period plot used for a period search (top panel) and the same plot but zoomed around the best solution with the lowest 휒2 value (bottom panel) 40 80 120 160 −160 −160 −120 −80 −40 40 80 120 160 −160 −160 −120 −80 −40 −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='64 χ2 Longitude Latitude (190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='44, 34.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 Rotational phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 Rotational phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 Residuals RMS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='055 Figure 3: 2017 SL16 - 휒2-map of all 312 solutions described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 and the solution with the lowest 휒2 value marked with a red cross (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The middle and bottom panels show the corresponding model fitting to the observational data and residuals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' rotational pole may we obtained from the 휒2 map, in this case the pole solution is more likely to be far from the eclip- tic than close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The 푅푀푆 for the entire dataset and for the model light curve with the lowest 휒2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 2016 CA138 2016 CA138 (H=23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3) is an Aten asteroid and one of the faintest objects in our sample, with a diameter of D=75 m assuming a visual geometric albedo pV=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 4 of 11 Earth co-orbitals properties No taxonomic or colour information is available for this object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Our solution for the rotational period (see Figure 4) is P=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3137 hrs (ΔP=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0016 hrs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The 휒2-map of the pole solutions is presented in the top panel of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' It indicates that the asteroid rotational pole should be close to the ecliptic pole but with two exceptions the vertical trenches along 60 and 260 degrees longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The solution with the deepest 휒2 minimum is at pole co- ordinates 휆1=307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='42◦ and 훽1=-83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='01◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' However, any other solution inside the trenches - which are almost 180 degrees apart in longitude and could be interpreted as mirror solu- tions - are plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The 푅푀푆 for the entire dataset and the model light curve with the lowest 휒2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Objects with previous period solutions Here we are presenting improved rotational state solu- tions for objects with previous spin rates published in Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' (418849) 2008 WM64 (418849) 2008 WM64 (H=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6) is an Apollo asteroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' It has a previously published rotation period of P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='02 h (Rowe, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Warner, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We combined our sample of dense and sparse data together with dense data from Rowe consisting of 21 light curves in ’R’ band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We note that these 2 4 6 8 10 Probe Period, hours 2 3 4 5 6 7 8 χ2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='5 Probe Period, hours 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='5 χ2 Figure 4: 2016 CA138 - 휒2 vs probe period plot used for a period search (top panel) and the same plot but zoomed around the solution with the lowest 휒2 value (bottom panel) 40 80 120 160 −160 −160 −120 −80 −40 40 80 120 160 −160 −160 −120 −80 −40 −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='94 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='59 χ2 Longitude Latitude (307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='42,−83.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 Rotational phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 Residuals RMS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='041 RMS1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='039 RMS2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='043 Figure 5: 2016 CA138 - 휒2-map of all 312 solutions (top panel) described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 and the best solution with the lowest 휒2 value marked with a red cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The middle and bottom panels show the corresponding model fitted to the observa- tional data and the fit residuals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Red and green colours represent the two different dates of observations - 17 and 18 February 2020, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' latter data do not improve the PABL distribution of the ob- servations as they were performed over a single night 4 days earlier than ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We then carry out a fit as for the asteroids using this combined dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Our rotational period search, presented in Figure 6, yields an asteroid spin rate of P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4077 hrs (ΔP=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0001 hrs), slightly larger than in our previous work (Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2021, P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='356±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='033 hrs) G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 5 of 11 Earth co-orbitals properties but closer to the solution obtained by Rowe (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The 휒2- map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 7 indicates a pole direction south of the ecliptic with formal best estimate at 휆1=217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='34◦ and 훽1=-39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='17◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The 푅푀푆 for the entire dataset and the model light curve with the lowest 휒2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' (138175) 2000 EE104 (138175) 2000 EE104 (H=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4) is an Apollo asteroid suspected to have debris spread along its orbit due to detec- tion of interplanetary magnetic field disturbances near the orbital nodes (Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' argue that this material can be boulders with diameters larger than 10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' (Jewitt, 2020) finds no evidence for co-moving companions or a dust particle trail and report a B-V colour of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='04 which they interpret as intermediate between C-class and S- class asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The mean B-R colour is consistent with that measured for Jovian Trojan and D-type asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Asteroids can shed material from their surface and onto heliocentric orbit if they rotate once every few hours or faster (Pravec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Jacobson and Scheeres, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Our in- vestigation shows that this asteroid is a relatively slow rotator with a period of about 14 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Our rotational period and pole solution estimates are presented in Figure 8 and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' A notable feature in this Figure is a lower 휒2 for pole solutions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='45 Probe Period, hours 5 10 15 20 25 30 χ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='404 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='406 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='408 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='410 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='412 Probe Period, hours 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 χ2 Figure 6: (418849) 2008 WM64 - 휒2 vs probe period plot used for a period search (top panel) and the same plot but zoomed around the best solution with the lowest 휒2 value (bottom panel) 40 80 120 160 −160 −160 −120 −80 −40 40 80 120 160 −160 −160 −120 −80 −40 −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='92 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='72 χ2 Longitude Latitude (217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='34,−39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 Rotational phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 Relative magnitude Per= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4077h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 Residuals RMS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='023 Figure 7: (418849) 2008 WM64 - 휒2-map of all 312 solutions described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 and the best solution with the lowest 휒2 value marked with a red cross (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The middle and the bottom panels show the corresponding model fit to the observational data and residuals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' north of the ecliptic and a single, broad minimum at pole coordinates 휆1=233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='86◦ and 훽1=68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='71◦ with corresponding period P1=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='9476 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' However, we cannot dismiss the two mirror solutions with similar 휒2 values south of the ecliptic, also well defined as 휒2 minima in Figure 9 and presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The 푅푀푆 for the entire dataset and the model light curve with the lowest 휒2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 6 of 11 Earth co-orbitals properties 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Comparison to NEAs In Figure 10 we compare the bulk properties of the four asteroids in our primary sample (filled circles) to NEAs en- tries in the LCDB database (Warner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' and Harris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' and Pravec, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2021, black points) including confirmed bi- nary systems (blue symbols) and objects in non-principal- axis or “tumbling” rotational state (green symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Our inferred sizes of sample asteroids are averages of two es- timates assuming typical albedo values for C- and S-type asteroids (푝퐶 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='057 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='013 and 푝푆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='197 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='051;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Pravec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The vertical lines in the figure indicate YORP-induced spinup timescales 휏푌 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1 and 1 Myr resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' using the expression 휏푌 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='68퐷2 Myr from Jacob- son (2014) appropriate for NEAs, where 퐷 is the diameter in km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' For a prograde rotator, we expect the YORP torque to evolve the asteroid towards a critically-spinning configu- ration in <1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In order to increase the sample size, we have included five Earth co-orbital asteroids from Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' (2021): (468909) 2014 KZ44, (468910) 2014 KQ76, (512245) 2016 AU8, (522684) 2016 JP and 2018 EB (open circles) where available data allows to estimate the spin period but not pro- duce a full rotational state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We then refer to these nine aster- oids as the extended sample to distinguish from the original 6 8 10 12 14 16 18 Probe Period, hours 3 4 5 6 χ2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='85 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='90 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='95 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='00 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='05 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='10 Probe Period, hours 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 χ2 Figure 8: (138175) 2000 EE104 - 휒2 vs probe period plot used for a period search (top panel) and the same plot but zoomed around the solution with the lowest 휒2 value (bottom panel) 40 80 120 160 −160 −160 −120 −80 −40 40 80 120 160 −160 −160 −120 −80 −40 −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='54 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='64 χ2 Longitude Latitude (233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='86, 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='71) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 and the best solution with the lowest 휒2 value marked with a red cross (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The middle and the bottom panels show the corresponding model fitting to the observational data and residuals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Red, green and blue colours represent the three different dates of observations 09 November 2018, 01 January 2019 and 02 January 2020, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' sample of four asteroids.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='45168 aSolution number (m - if it is a mirror solution) bThe difference between the minimum and the maximum of the data cThe amplitude of the data computed as a difference between the five point average around the minimum and the maximum of the data dThe amplitude of the modeled light curve eThe dark facet area in % would begin to come apart (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Harris and Pravec, 2005, and references therein), a regime populated by the smallest primaries within the binary asteroid population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Arguably, the most noteworthy case is that of 2017 SL16, with an estimated 퐷 = 20−36 m the smallest asteroid in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Its fast, ∼15 minute rotation period suggests an as- teroid held together by internal strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The next smallest object is 2016 CA138 (퐷 = 66 − 121 m) with a spin pe- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1 1 10 100 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1 1 Rotation Frequency (d-1) Diameter (km) All NEOs Binary NEOs Tumbling NEOs Borisov et al 2021 This work EE104 CA138 SL16 WM64 1Myr 100kyr 10kyr Figure 10: Size vs spin rate of co-orbital asteroids in our sam- ple (red points) compared to NEAs entries in the Asteroid Lightcurve Data Base (LCDB Bundle v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0) as of December 2021, including confirmed binary and tumbling asteroids (blue and green points resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The horizontal line corresponds to the critical spin rate 휔푐푟푖푡(휌) for 휌 = 2000 kg m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Different YORP spinup times for the asteroids are shown as vertical red lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' riod of ∼5 hr, rather slow but still within the observed range for objects of similar size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' A more extreme, though by no means exceptional, case is the ∼14 hr period of 2000 EE104, (퐷 = 255 − 470 m), slower than most asteroids in the same size range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' This asteroid, along with 2016 CA138 and 2017 SL16, is located in the regime occupied by tumbling aster- oids although we note here that no evidence of a secondary period or the presence of satellites was found in our photo- metric data for these asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Numerical Simulations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Simulation setup To investigate the orbit evolution of the asteroids, we used the HYBRID symplectic state propagation scheme avail- able in the MERCURY package (Chambers, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' This scheme accurately models close encounters between test par- ticles and planets by switching from mixed-variable sym- plectic to Bulirsch-Stoer state propagation within a certain distance from a planet, set to 2 Hill radii for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The solar system model included the eight major planets from Mercury to Neptune and was strictly Newtonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Initial plan- etary state vectors were retrieved from the HORIZONS on- line ephemeris service (Giorgini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 1996) at the J2000 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Each asteroid was cloned 20 times with starting con- ditions for each clone generated by applying the linear trans- formation 퐲 = 퐏횲1∕2퐱 (2) to a 6-dimensional normally-distributed random variate 퐱 (Duddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Here, 퐏 and 횲 contain the eigenvec- G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 8 of 11 Earth co-orbitals properties Table 4 Summary results of the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The four asteroids in our primary sample are indicated in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 1−휎 Dispersiona Asteroid 푎 퐼 Co-orbital for 푎 (au)/푒/퐼 (deg) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 3 (Number) Designation (au) 푒 (◦) mode @−1Myr @+1Myr (138175) 2000 EE104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='24/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='14/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='30/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='14/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='31 (418849) 2008 WM64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='11 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='5 Passing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='06/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='07/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='03/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='07/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='08 2016 CA138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='05 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='7 Horseshoe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='18/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='13/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='42 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='8 N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='32/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='08/3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='33 (512245) 2016 AU8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='20 9.' 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='15/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='09/6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='17 2018 EB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='01 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4 N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='06/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='06/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='08 aDispersions for the five objects not in our primary sample are averages from the forward and backward runs tors and eigenvalues of the asteroids’ state covariance ma- trices, retrieved from the Near-Earth Objects Dynamic site for the epoch JD2459200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='5 = 2020 December 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Both massive bodies and test particles were integrated to the same epoch before the start of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' One question to be asked here is whether including the size-dependent Yarkovsky drag force in our dynamical model might change the outcome in a significant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Fenucci and Novaković (2020) investigated this question for the Earth quasi-satellite (469219) Kamo’oalewa, an object compara- ble in both size and orbit to the smallest object in our sam- ple, 2017 SL16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Though Yarkovsky does change the orbital evolution of Kamo’oalewa over millions of yr and its resi- dence time as an Earth co-orbital, actual differences from the gravity-only case were quite small and the overall effect on the evolution of the orbit was not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' For this reason, and to minimise the computational overhead of our runs, we decided not to include the Yarkovsky effect in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Two simulation batches were run for the four groups of asteroid clones plus the nominal orbits, one for 104 yr and the other for 106 yr, backwards and forwards from the starting epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The integration step size in both cases was 4 days, the output step was 10 yr for the 104 yr runs and 103 yr for the 106 yr runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Results Orbital evolution of the clone ensembles for each aster- oid is presented in Figure 11 where we also show the running mean and standard deviation for each clone ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The standard deviation for the three actions 푎, 푒 and 퐼 relative to the J2000 ecliptic at 푡 = ±106 yr is separately reported in Table 4 and serves as a measure of orbital stability for each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' It is worth noting at this point that, according to our definition of stability, orbit transitions between coor- bital modes while in the 1:1 resonance are considered stable since the semimajor axis continues to oscillate around 1 au while 푒 and 퐼 do not diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Below we discuss each asteroid separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' (138175) 2000 EE104 This asteroid has the most unstable orbit of those in- vestigated in this work, with 휎푎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='05 au after ±104 yr and ≳0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='20 au after ±106 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' This is probably due to its moderate eccentricity and low inclination, allowing frequent and rela- tively slow encounters with Venus as well as the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' None of the clones remains within the Earth’s co-orbital region (|푎 − 1 au| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='01 au) for more than a few hundred years from the start of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' (418849) 2008 WM64 This asteroid is slowly drifting backwards with respect to the Earth in what we refer to as a passing orbit (Namouni, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Our simulations show that all orbits trace identical paths in 푎, 푒 and 퐼 for at least 104 yr in the past and in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Longer-term, the asteroid has likely been in a pass- ing orbit for the past 2×105 yr while the future evolution of the orbit is less certain, with the clone semimajor axes be- ginning to disperse after a few times 104 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The orbit dis- persion 106 yr from the present is 휎푎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='06 au forwards and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='03 au backwards, with five clones remaining within the co- orbital region until 푡 = ±106 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Note the anti-correlated oscillations of 푒 and 퐼, indicative of the Kozai regime with the argument of perihelion 휔 librating around 180◦, typical of moderate-퐼 asteroids in the vicinity of the Earth’s orbit (Michel and Thomas, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Namouni, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' This state per- sists for at least 5×105 yr in the past and in the future and, together with the moderate-to-high inclination (퐼 ≃ 34◦), offers some protection against the orbit-changing effects of planetary encounters (Michel and Thomas, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 2016 CA138 This asteroid is currently in an Earth horseshoe orbit, qualifying therefore as the 13th Earth horseshoe (Kaplan and Cengiz, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Figure 12 shows that the resonant angle Δ휆 = 휆−휆Earth for the clone orbits currently librates around Δ휆 = 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' All orbits enter the horseshoe phase ∼6×103 yr before 푡 = 0 and exit it ∼600 yr after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We also note a short QS episode between 푡 = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 × 103 and 푡 = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 × 103 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In the forward runs, knowledge of the future orbital state begins G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 9 of 11 Earth co-orbitals properties to degrade after ∼103 yr, however all orbits continue within some co-orbital mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In our 1 Myr runs, we find that con- finement of the asteroid’s orbit within the Earth’s co-orbital region persists for several times 104 yr in the past and in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Interestingly, the final 푎 dispersion is higher in the backwards integrations (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='15 au) compared to the forward runs (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='04 au) with 1 clone still in the co-orbital region at 푡 = +1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Note also the rapid increase of the eccentricity from initially ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='05 at 푡 = 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='20 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='30 after ∼2×105 yr, suggesting that the co-orbital resonance helps to keep 푒 be- low ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='15 in the few tens of thousands of years closest to 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 2017 SL16 The orbital evolution of this asteroid was recently inves- tigated by Kaplan and Cengiz (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Those authors showed that SL16 is currently in an QS-HS asymmetric horseshoe configuration, having transitioned into this state from a pass- ing orbit ∼100 yr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In addition to confirming the present QS-HS state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 12), our simulations of the asteroid’s or- bital evolution up to 104 yr from the present are in very good agreement with Kaplan and Cengiz (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' However, whereas those authors showed the asteroid to evolve to a higher 푒 and 퐼 orbit in the future, our forward integrations show a differ- ent outcome, where 푒 and 퐼 remain approximately constant from about 푡 = +3 × 103 yr until the end of the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We attribute this to the improving orbit knowledge for this as- teroid, with a better-determined orbit being available to us than in the Kaplan and Cengiz work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In the longer-term, the ensemble behaviour of 푒 and 퐼 remains unchanged while the orbital semimajor axis disperses by as much as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2 au, with 3 of the 21 orbits still in the co-orbital region at 푡 = +1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Overall dynamical properties and relation to rotational state Here we combine the statistical dispersions for each or- bital action in Table 4 into an ensemble indicator of orbital stability for each asteroid, defined as 휎2 = (휎푎∕푎)2 + 휎2 푒 + (휎퐼∕퐼)2 (3) For this purpose, we carried out additional numerical simulations for the remaining five objects in the extended sample as for our original sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' None of these additional objects are currently locked in resonance, yet their orbital stability, as quantified by 휎, are similar to those of the pri- mary sample (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The overall stability of these nine asteroids should therefore be representative of the popula- tion of NEAs with 푎 ∼1 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We find that the least stable objects in the extended sam- ple are 138175 and 468910 (휎 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3) while the most stable (휎 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1) are 418849 and 2018 EB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The presence of an as- teroid in the co-orbital resonance appears, therefore, to be a poor indicator of dynamical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Instead, the individ- ual dispersions reported in Table 4 are correlated with the orbital actions, so that low 푒 and high 퐼 orbits are the most stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' This suggests that the principal cause of instability is orbit-changing planetary encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Indeed, a high orbit in- clination helps to avoid frequent close encounters with plan- ets and moderates their orbit-changing effects even when they do occur (Michel and Thomas, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Furthermore, an asteroid in an orbit with 푒 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='25 may approach Venus as well as the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' On the other hand, resonance-locking episodes and passing orbit states typically last ∼104 yr (Fig- ure 11, see also Morais and Morbidelli (2002)) and much shorter than a Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In this narrative, 138175 diffuses the fastest due to its high eccentricity allowing close encounters with Venus that disrupt the co-orbital resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In the same way, the mod- erate orbital diffusion displayed for 2016 CA138 and 2017 SL16 is caused by deterministic changes while in the 1:1 resonance with Earth and intermittent forays of the orbital eccentricity above the Venus-crossing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Finally, the low-푒, high-퐼 orbits of 418849 and 2018 EB mitigate against close approaches to both Earth and Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We return now to the question posed in the Introduction, namely whether orbital stability and the rotational state of NEAs at 1 au are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' At first glance, this does not appear to be the case since, for example, the set of six asteroids near the spin rate barrier (Figure 11) is a mixture of stable and unstable cases (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Alternatively, we can choose to consider only those as- teroids at the two extremes of the stability spectrum, ie those with the highest and lowest values of 휎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' If a relation of the type we are searching for existed, we would expect the rota- tional properties of those two groups to differ the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Here we find see that the spin rates of 418849, 468910 and 2018 EB are near-critical while that of 138175 is sub- critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' One might regard this as evidence that the vari- able dynamical environment of 138175 interferes with the spinning-up action of the YORP torque, preventing it from reaching a near-critical spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' In a sample of only four, however, this evidence is far from conclusive and, while such a relationship might still exist, extending our study to a much larger sample of asteroids appears necessary to eke it out of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Conclusions We determine the rotational periods and axis orienta- tions of four Earth co-orbital asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The smallest object - 2017 SL16 with a diameter of a few tens of meters - has a rotational period P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3188 hrs which suggests that it is monolithic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Even though we can- not find a definitive pole solution, we can say that it is likely not near the ecliptic and the best solution from the five mod- els with similar 휒2 values has coordinates 휆1=190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='44◦ and 훽1=34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='32◦ The other extreme case is the asteroid (138175) 2000 EE104, which we find to be a slow rotator with a spin rate of P=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='9476 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The pole position is also not uniquely deter- mined but most probably it lies above the ecliptic with the formal best-fit pole at 휆1=233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='86◦ and 훽1=68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='71◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 10 of 11 Earth co-orbitals properties The object with the best quality of observations and re- spectively the best results is (418849) 2008 WM64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' It has only one clear definitive solution for the rotational period: P=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='4077 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Also, the pole solution lies on a single spot of the 휒2 search map at coordinates 휆1=217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='34◦ and 훽1=-39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='17◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' For the second smallest object in our sample - 2016 CA138 with an estimated diameter of 75 m - we determine the spin rate to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3137 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The 휒2 map shows two vertical stripes with low 휒2 values at longitudes 180◦ apart - around 80◦ and 260◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We compared the rotational properties and sizes of our asteroid sample to NEA entries in the LCDB database (Warner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' and Harris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' and Pravec, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Our overall conclusion is that the size vs rotational frequency distribu- tion of the co-orbital asteroids appears not to differ from the one of the NEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We made numerical simulations in order to investigate if there is a relation between the orbit stability and the ro- tational state of Earth co-orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Our results show that the co-orbital resonance is not affecting the orbit stability, but the orbit itself is responsible for that and mainly its eccen- tricity (푒) and inclination (퐼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Orbits with low 푒 and high 퐼 are the most stable because firstly highly-inclined orbits are relatively stable against close encounters with planets and secondly orbits with high 푒 may approach Venus as well as the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' We cannot make a definitive conclusion if the orbit sta- bility and rotational state of the asteroids are related, so we need further investigations and observations to increase our sample in order to obtain more statistically significant re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Acknowledgements This work was supported via grant ST/R000573/1 from the UK Science and Technology Facilities Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The au- thors gratefully acknowledge observing grant support from the Institute of Astronomy and National Astronomical Ob- servatory, Bulgarian Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Astronomical research at the Armagh Observatory & Planetarium is grant- aided by the Northern Ireland Department for Communities (DfC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The authors also acknowledge DfC for the FoReRo2 instrument development contribution from the Armagh Ob- servatory & Planetarium toward the new CCD camera Andor iKon-L used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' References Borisov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Christou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Bagnulo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Cellino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Lightcurve and spin rates of Earth co-orbital asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Minor Planet Bulletin 48, 268–271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Chambers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' A hybrid symplectic integrator that permits close encounters between massive bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' MNRAS 304, 793–799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 1046/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1365-8711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='02379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Christou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' A numerical survey of transient co-orbitals of the terrestrial planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 144, 1–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Di Ruzza, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Pousse, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Alessi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' On the co-orbital asteroids in the solar system: medium-term timescale analysis of the quasi-coplanar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 390, id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 115330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Duddy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Lowry, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Wolters, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Christou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Weissman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Green, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Rozitis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Physical and dynamical characteri- sation of the unbound asteroid pair 7343-154634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' A&A 539, A36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1051/0004-6361/201118302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Durech, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Kaasalainen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Warner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Fauerbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Marks, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Fauvaud, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Fauvaud, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Vugnon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Pilcher, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Bernasconi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Behrend, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Asteroid models from combined sparse and dense photometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' A&A 493, 291–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1051/0004-6361: 200810393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Durech, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Sidorin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Kaasalainen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' DAMIT: a database of as- teroid models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' A&A 513, A46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1051/0004-6361/200912693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Fenucci, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Novaković, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The role of the Yarkovsky Effect in the long-term dynamics of asteroid (469219) Kamo’oalewa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 162, id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3847/1538-3881/ac2902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Giorgini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Yeomans, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Chamberlin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Chodas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Jacobson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Keesey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Lieske, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Ostro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Standish, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Wimberly, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' JPL’s on-Line Solar System data service, in: AAS/Division for planetary sciences meeting abstracts #28, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Harris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Pravec, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Rotational properties of asteroids, comets and TNOs, in: Symposium S229: Asteroids, Comets, Meteors, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 439– 447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Jacobson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Small asteroid system evolution, in: Complex Plan- etary Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 108–117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Jacobson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Scheeres, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Dynamics of rotationally fissioned asteroids: Source of observed small asteroid systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 214, 161– 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='icarus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='009, arXiv:1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Jewitt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 138175 (2000 EE104) and the source of interplanetary field enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The Planetary Science Journal 1, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='3847/ PSJ/aba68f, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='07192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Kaasalainen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Torppa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Muinonen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Optimization meth- ods for asteroid lightcurve inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The complete inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 153, 37–51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1006/icar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Kaplan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Cengiz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Horseshoe co-orbitals of Earth: current pop- ulation and new candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' MNRAS 496, 4420–4432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1093/ mnras/staa1873, arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='14451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Lai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Russell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Wei, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Connors, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Delzanno, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Possible potentially threatening co-orbiting material of aster- oid 2000EE104 identified through interplanetary magnetic field distur- bances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Meteoritics and Planetary Science 52, 1125–1132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1111/ maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='12854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Michel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Thomas, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The Kozai resonance for near-Earth asteroids with semimajor axes smaller than 2AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' A&A 307, 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Morais, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Morbidelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The population of near-Earth aster- oids in coorbital motion with the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 160, 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1006/ icar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Namouni, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Secular interactions of coorbiting objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 137, 293–314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1006/icar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='6032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Pravec, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Harris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Kušnir ak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Galád, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Hornoch, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Abso- lute magnitudes of asteroids and a revision of asteroid albedo estimates from WISE thermal observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 221, 365–387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Pravec, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Vokrouhlický, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Polishook, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Scheeres, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Harris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Galád, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Vaduvescu, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Pozo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Barr, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Longa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Vachier, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Colas, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Pray, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Pollock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Reichart, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Ivarsen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Haislip, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Lacluyze, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Kušnirák, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Henych, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Marchis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Macomber, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Ja- cobson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Krugly, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Sergeev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Leroy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Forma- tion of asteroid pairs by rotational fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Nature 466, 1085–1088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1038/nature09315, arXiv:1009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Qi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Qiao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Stability analysis of Earth co-orbital objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' As- tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 163, id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Rowe, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Lightcurve analysis of 6 asteroids from RMS observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Minor Planet Bulletin 45, 292–294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Rubincam, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Radiative spin-up and spin-down of small asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 148, 2–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Scheeres, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Marzari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Rossi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Evolution of NEO rotation rates due to close encounters with Earth and Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 170, 312– G.' metadata={'source': 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and inclination 퐼 relative to the ecliptic plane for the nominal orbit and 20 clones of each asteroid over 104 yr (left) and 106 yr (right) from 푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Black dashed and solid lines represent the mean and standard deviation for each clone set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The red plus symbol indicates the starting orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='Page 12 of 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='Earth co-orbitals properties ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='Asteroid 2016 CA138 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1000-800 -600 -400 -200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='200 400 600 800 1000 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='Resonant angle (degrees) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='Figure 12: Evolution of the resonant angle Δ휆 = 휆 − 휆Earth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='for the clones of Earth co-orbitals 2016 CA138 (top) and 2017 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='SL16 (bottom) and for 103 yr backwards and forwards from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='the present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The dashed horizontal line and red plus symbol indicate the Δ휆 = 0◦ datum and the starting orbit respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Tricarico, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' The near-Earth asteroid population from two decades of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 284, 416–423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='icarus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='008, arXiv:1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='06328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Walsh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Richardson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' A binary near-Earth asteroid forma- tion: Rubble pile model of tidal disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Icarus 180, 201–216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Walsh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Richardson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Michel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Rotational breakup as the origin of small binary asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Nature 454, 188–191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Warner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Near-Earth asteroid lightcurve analysis at CS3-Palmer Divide station: 2017 October-December.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Minor Planet Bulletin 45, 138– 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Warner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' and Harris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' and Pravec, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Asteroid lightcurve database (LCDB) bundle V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' NASA planetary data system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' URL: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='26033/j3xc-3359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Wiegert, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Innanen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', Mikkola, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=', 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' An asteroidal companion to the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Nature 387, 685–686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' Borisov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 13 of 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE1T4oBgHgl3EQfqwVD/content/2301.03346v1.pdf'} diff --git a/GtE1T4oBgHgl3EQfFQMH/content/2301.02899v1.pdf 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Despite their success, these models are still vulnerable +regarding small perturbations, which can be used to craft the so-called adversarial examples. +Different approaches have been proposed to circumvent their vulnerability, including formal +verification systems, which employ a variety of techniques, including reachability, optimization +and search procedures, to verify that the model satisfies some property. In this paper we propose +three novel reachability algorithms for verifying deep neural networks with ReLU activations. +The first and third algorithms compute an over-approximation for the reachable set, whereas the +second one computes the exact reachable set. Differently from previously proposed approaches, +our algorithms take as input a V-polytope. Our experiments on the ACAS Xu problem show that +the Exact Polytope Network Mapping (EPNM) reachability algorithm proposed in this work +surpass the state-of-the-art results from the literature, specially in relation to other reachability +methods. +1 +Introduction +Regardless of the success of deep neural networks in computer vision and natural language +processing, these models are susceptible to small perturbations applied to their inputs, i.e. it is +possible to misguide the model output by applying a designed perturbation to a given input. For +instance, the ACAS Xu model [1] (explained later in detail), that responded differently from expected +while under specific circumstances. The inputs purposefully designed to force a misbehavior of the +neural network are denoted as adversarial examples [2]. +To overcome such vulnerability many different approaches have been previously employed. One +proposed the application of algorithms that were able to generate adversarial examples, the so called +adversarial attacks, and subsequently applied these inputs in the training process of the network +[2, 3, 4, 5]. There were also those approaches that aim to identify the adversarial examples before +feeding them as input to the neural network [6, 7, 8, 9]. Even though these procedures helped to +reduce the vulnerability of the neural networks, these models remained vulnerable to adversarial +attacks. +∗joao.zago@posgrad.ufsc.br +†eduardo.camponogara@ufsc.br +‡eric.antonelo@ufsc.br +1 +arXiv:2301.12001v1 [cs.LG] 27 Jan 2023 + +Formal methods were also applied to certify or guarantee that the model behaves as expected +under some circumstances or within a specified domain region, nevertheless [10] showed that the +verification problem is NP-hard, leaving the process of certifying large models still an open problem. +The existing formal procedures can be classified into three different categories: 1) reachability +methods; 2) optimization methods; and 3) search methods. The first one relies on calculating the +output mapping of an input set [11, 12, 13], the second one comprises the application of math- +ematical optimization (Mixed Integer-Linear Programming or Convex Optimization) to identify +counter-examples [14, 15, 16, 17], and the third makes use of both reachability and optimization +approaches in conjunction with search methods for identifying counter-examples [10, 18]. +In this paper, we propose three novel reachability algorithms: APNM and PAPNM algorithms +that compute an over-approximation for the output, while EPNM which computes the exact map- +ping. +We present demonstrations on the behavior and correctness of these algorithms and case +studies of their applications for comparison with existing algorithms from the literature. We also +show that the algorithms proposed in this work are highly parallelizable. +The rest of this paper is organized into five sections: Section 2 gives an overview of the existing +algorithms and related works; Section 3 describes the proposed algorithms; Section 4 addresses the +demonstrations regarding the completeness and soundness of the proposed algorithms; Section 5 +presents a study case for application and comparison; and finally Section 6 discusses the outcome of +the procedures presented in this work. +2 +Related Works +Formal verification of neural networks is receiving a huge amount of attention mainly because of +its importance in security sensitive tasks such as the application of neural networks in autonomous +vehicles, systems controllers, aeronautics, and several other applications that can possibly involve +financial, human or environmental injury [18, 1]. +As previously presented, there are three main research areas developed to provide guarantees for +neural networks: 1) reachability methods; 2) optimization methods; and 3) search methods. In this +work, we propose three novel approaches regarding reachability analysis, which consist of two major +steps: computing the output set by mapping a subset of the neural network domain and comparing +the output set with the specification. +One of the algorithms developed in the literature to verify neural networks by means of reach- +ability analysis is denoted as MaxSens [12], which computes an over-approximation to the output +reachable set and compares it to the desired specification. As this algorithm computes an over- +approximation it does not satisfy the completeness property of a verification algorithm, although it +is sound (if the property is verified then the algorithm guarantees that it is satisfied). To compute +such an over-approximation, MaxSens divides the input set into several smaller hyperrectangles and +computes the maximum sensitivity for each of them layer-by-layer. The output of this algorithm +consists of the union of several hyperrectangles, which approximates the exact reachable set. Note +that their approach can be applied to neural networks with any activation function. +Another approach from the literature, denoted as ExactReach [11], computes the exact reachable +set given an input set. This procedure takes as inputs a convex H-polytope (a polytope represented +by its inequalities) and computes the exact mapping layer-by-layer. +The authors proposed this +approach specifically for the ReLU activation function, which is reasonable as this function has +achieved promising results for convolutional and fully connected neural networks, which are widely +applied in different applications. Due to the nature of the ReLU activation function, the authors +separated the non-linear mapping process in three cases: 1) all the elements of the input are positive; +2 + +2) all the elements of the input are negative; and 3) the input has positive, negative or null elements. +These cases cover all the mapping possibilities regarding the ReLU activation. +Similarly to the aforementioned approaches, we propose in this paper three novel algorithms for +reachability analysis. However, instead of using the H-polytope representation, each of our proposed +procedures take as input set a V-polytope (a polytope represented by its vertices). We demonstrate +the correctness of both algorithms and compare them with the literature (not only with those that +make use of reachability analysis) by using a case study (ACAS Xu [1]). +3 +Algorithms +In this section we will state the verification problem and the three algorithms proposed in this +work. +The demonstrations regarding the correctness of each procedure will be presented in the +following section. +3.1 +Problem statement +Let F : Rn �→ Rm represent a mapping given by a neural network composed of L layers. Then, +for a given x ∈ Rn, F(x) = (FL ◦ FL−1 ◦ · · · ◦ F2 ◦ F1)(x), where Fl is the mapping given by the +l-th layer. Further, Fl consists of the composition of an affine mapping and a non-linear mapping +(in this case a ReLU function): Fl(xl) = ReLU(Wlxl + θl), where Wl and θl denote the weight +matrix and bias vector of the l-th layer, respectively, and xl is the input to the same layer. +Suppose that X ⊆ Rn is the input set that we want to verify and R = {F(x) | x ∈ X} is the +exact output set associated with X, which resulted from the application of the neural network F +to each input in X. Moreover, the set Y comprises the expected output set for the inputs from X. +Then, the verification problem consists of assuring that: +R ∩ ¬Y = ∅. +(1) +In other words, we want to guarantee that there will not exist an input of F in X that will cause +the network to generate an undesired output (an output that is not in Y). +To perform such a verification, one needs to start by calculating the output set associated with +X. As we presented in previous sections, there are approaches that compute the exact reachable set +and other approaches that over-approximate it. For those that compute an approximation for the +output set, denoted by �R, we will have that R ⊆ �R. Then, we can see that if �R ∩ ¬Y = ∅, then the +property given by the Equation 1 will still be assured. In the following sections we will present the +proposed algorithms. +3.2 +Approximate Polytope Network Mapping (APNM) +The first algorithm proposed in this work is called Approximate Polytope Network Mapping +(APNM) and consists of a procedure to compute an over-approximation for the reachable set. Let +P be a convex closed polytope, defined as a convex combination of its vertices, namely +P = +� o +� +i=1 +λivi +���� +o +� +i=1 +λi = 1, λi ≥ 0 and vi ∈ V, ∀i ∈ {1, . . . , o} +� +, +(2) +where V is the set of vertices of P (V = {v1, . . . , vo}). To achieve its goal, the algorithm has five +parts: +3 + +1. Affine map of the vertices with weights and biases; +2. Adjacent vertex identification; +3. Polytope intersection with an orthant’s hyperplanes; +4. ReLU mapping; and +5. Removing non-vertices. +In what follows, the way the algorithm works will be explained with a simple 2-dimensional +problem for visualization purposes. The input set P is presented in Figure 1: the set of vertices is +V = {v1, v2, v3, v4}, where v1 = (1.0, 1.0), v2 = (−1.0, 1.0), v3 = (−1.0, −1.0) and v4 = (1.0, −1.0); +and the set of edges is E = {(v1, v2), (v1, v4), (v2, v3), (v3, v4)}. +Figure 1: The input set for visualizing each step of the algorithm. +Affine map of the vertices with weights and biases +The affine map comprises the product of each vertex of P by the weights associated with layer l +plus the biases of the same layer. Algorithm 1 contains the steps to compute the affine map of all +vertices from V. +Algorithm 1: Affine Map (AM) +Input: V ∈ Ro×n, V = [v[1], . . . , v[o]], v[i] ∈ Rn , ∀i ∈ {1, . . . , o}, W ∈ Rm×n and θ ∈ Rm +Function AM(V, W, θ): +let Z[1..o] be a new array, where each Z[i] is an empty array, ∀i ∈ {1, . . . , o} +for i ∈ {1, . . . , o} do +Z[i] = Wv[i] + θ ; +end +return Z +4 + +2 +V2 +V1 +1 +0 +-1 +-2 +-2 +-1 +0 +1 +2 +1By applying the affine map to the input set, presented in Figure 1, we have the output of this +step as shown in Figure 1. The weight matrix W and biases θ employed in this toy example are as +follows: +W = +�0.492693 +−1.29232 +0.925861 +0.675146 +� +, +θ = +�−0.18857972 +−0.14839205 +� +(3) +Note that, as expected, the output of the current operation is a simple affine transformation of the +vertices of P. +Figure 2: Representation of the output of the affine map operation. +Adjacent vertex identification +Following the affine map, the edge identification process takes place. Let vi, vj ∈ V be two +distinct vertices, such that i ̸= j. If there is at least one combination of the vertices of V, except +from vi or vj, that allow us to compute the middle point of vi and vj, given by (vi + vj)/2, then +vi and vj are not adjacent. Equation 4 presents this feasibility problem as a MILP (Mixed Integer- +Linear Programming), where λ ∈ Ro is the vector used to represent the convex combination of the +5 + +2 +V1 +1 +0 +-1 +V3 +-2 +-2 +-1 +0 +1 +2 +1vertices of P and η ∈ {0, 1} is the binary variable that enables evaluating vi or vj separately. +max +0 , +(4a) +s.t. +(vi + vj) +2 += +o +� +k=1 +vkλk , +(4b) +o +� +k=1 +λk = 1 , +(4c) +λk ≥ 0 ,∀k ∈ {1, . . . , o} , +(4d) +λi ≤ η , +(4e) +λj ≤ 1 − η , +(4f) +η ∈ {0, 1} +(4g) +Algorithm 2 presents the identification of adjacent vertices. It creates an undirected graph using +an adjacency list as the data structure to represent the 1-skeleton of the polytope, which is the edge +structure of some polytope [19, 20]. We denote the adjacency list of the undirected graph by E, +were each element is a set containing the indices of the adjacent vertices. +Algorithm 2: Edge-skeleton Identification (EI) +Input: V ∈ Ro×n, V = [v[1], . . . , v[o]], v[i] ∈ Rn , ∀i ∈ {1, . . . , o} +Function EI(V): +let E[1 . . . o] be a new array, where each E[i] , ∀i ∈ {1, . . . , o}, corresponds to an empty +set +for i ∈ {1, . . . , o} do +for j ∈ {i + 1, . . . , o} do +if ¬∃λ ∈ Ro : 1 +2(v[i] + v[j]) = �o +k=1 v[k]λ[k] ∧ �o +k=1 λ[k] = 1 ∧ λ[k] ≥ 0 , ∀k ∈ +{1, . . . , o} ∧ (λ[i] = 0 ∨ λ[j] = 0) then +E[i] ← E[i] ∪ {j} +end +end +return E +Following the toy problem, where the affine map is shown in Figure 2, the undirected graph that +represents the edge structure of the affine map of P is presented in Figure 3. This graph G = (V, E) +consists of the vertex set V = {v1, v2, v3, v4} and edge set E = {(v1, v2), (v1, v4), (v2, v3), (v3, v4)}. +Notice that each edge in the undirected graph indicates that the associated vertices are connected +by an edge of the corresponding polytope. +Polytope intersection with orthant’s hyperplanes +The next step for the algorithm concerns the identification of those points at the intersection +of the hyperplanes that define each orthant with the edges of P. As the edges of the polytope +were previously computed, only the edges whose extreme points belong to different orthants need +6 + +Figure 3: Undirected graph that represents the edge structure of the output of the affine map of P. +to be identified, i.e., extreme points that have at least one component with a different sign. So, we +compute the difference of the sign for each element of both extreme points for a given edge, given +by Equation 5: +a = sign(vi) − sign(vj) +(5) +where i denotes the vertex under consideration, j ∈ Ei represents the index of the vertices that are +adjacent to vi, and sign : Rn → Rn is the signal mapping that associates 1 for the positive elements, +−1 for the negative elements, and 0 for the null ones. We also define a function σ : R → R, given by +Equation 6. For the cases where a component of a is greater than or equal to 2, we have that the +sign indeed changed. For those cases where the difference is equal to 1, there was a null component +in one of the vertices, which does not indicate that they are in different orthants. +σ(α) = +� +1, +if |α| ≥ 2 +0, +otherwise +(6) +Then, for those cases where there is a sign change, or, in other words, for those cases that satisfy +the condition: +n +� +k=1 +σ(ak) ̸= 0 +(7) +we compute those points at which the intersection between the edge and the hyperplane occurred. +Each sign change generates an intersection point. +Given two vertices vi and vj, suppose that there is a k ∈ {1, . . . , n} for which σ(ak) ̸= 0, +indicating that vi and vj belong to different orthants. +So, there exists a value of λ, such that +7 + +V.00 ≤ λ ≤ 1, defining a convex combination of the vertices that lies on the orthant hyperplane, namely +a λ that satisfies: +λvj,k + (1 − λ)vi,k = 0 ⇐⇒ λ = +−vi,k +vj,k − vi,k +(8) +Figure 4 contains a visual representation of this process for the case of vertices v1 = (−0.988207, 1.45261) +and v4 = (1.59643, 0.102323). Notice that a = sign(v1) − sign(v4) = (−1, 1) − (1, 1) = (−2, 0), +thus σ(a) = (1, 0) implying that v1 and v4 lie in different orthants, and the edge connecting them +intercepts the hyperplane defining the respective orthant at a point p. By using the above equation, +we compute the convex combination parameter λ to be 0.382338. +Figure 4: Representation of the intersection identification process. The blue dots represent two +adjacent vertices and the line segment is the edge connecting them. The red dot is the intersection +point of the edge with respect to the orthant hyperplane supporting plane. +Then, we compute each component s ∈ {1, . . . , n} of the point p ∈ Rn, as presented by Equation +9, +ps = (vj,s − vi,s)λ + vi,s, +(9) +which represents the intersection between an edge of P and one of the hyperplanes of an orthant from +Rn. For the example in Figure 4, the intercept of the edge (v1, v4) with the supporting hyperplane +{x = (x1, x2) : x1 = 0} of the orthant {x : x ≥ 0} is p = (0, 0.93635). +The procedure, formalized by Algorithm 3, computes the intersection points between each edge +of P and each orthant’s supporting hyperplane that are intercepted by the edge. +The result of the intersection identification for the output of the affine map of P is presented in +Figure 5. There we present all the vertices of the polytope in addition to the computed intersection +points. +ReLU Mapping +After calculating the intersection points between edges and orthant’s hyperplanes, the ReLU +mapping of the resulting points is computed. We apply the mapping ReLU : Rn → Rn, defined by +Equation 10, to all the points given by the previous procedure. Algorithm 4 summarizes the process +8 + +2 +1 +V +0 +-1 +-2 +-2 +-1 +0 +1 +2 +1Algorithm 3: Orthant intersection identification (II) +Input: V ∈ Ro×n where V = [v[1], . . . , v[o]], v[i] ∈ Rn , ∀i ∈ {1, . . . , o}, +E = [E[1], . . . , E[o]] +Function II(V, E): +for i ∈ {1, . . . , o} do +/* For each edge of vertice i +*/ +for j ∈ E[i] do +a ← sign(v[i]) − sign(v[j]) +for k ∈ {1, . . . , n} do +if σ(a[k]) ̸= 0 then +λ ← +−v[i,k] +v[j,k]−v[i,k] +let p be a point in Rn +for s ∈ {1, . . . , n} do +p[s] ← v[j,s]−v[i,s] +v[i,s] +λ +end +V ← [V | p] /* Append p to matrix V +*/ +end +end +end +return V +Figure 5: Intersection identification results representation. +where V is the output vertex set from Algorithm 3. +ReLU(x) = max(0, x) +(10) +We can see the result of the application of the ReLU mapping as presented in Figure 6. As +expected, all the vertices were projected to the positive orthant, whereby vertices v2, v5, and v6 are +projected onto the origin. Notice that the output set is not convex. +9 + +2 +V1 +1 +VA +V: +0 +NB +-1 +V6 +V3 +-2 +-2 +-1 +0 +1 +2 +1Algorithm 4: ReLU map +Input: V ∈ Ro×n, V = [v[1], . . . , v[o]], v[i] ∈ Rn , ∀i ∈ {1, . . . , o} +Function ReLU(V): +for i ∈ {1, . . . , o} do +for j ∈ {1, . . . , n} do +v[i, j] ← max(0, v[i, j]) +end +end +return V +Figure 6: Representation of the output of the ReLU mapping of the vertices and intersection points. +Removing Non-vertices +To simplify the output of a single layer, algorithm APNM computes the convex hull on the +output of the ReLU mapping. To compute the convex hull, the algorithm removes the points that +are not vertices. To create a generalized process for eliminating non-vertices, and due to the high +dimensionality of the internal layers of the neural network, we propose the application of a feasibility +analysis to identify the points that can be expressed as a convex combination from the remaining +points. +10 + +2 +V +1 +V +0 +V5 +V3 +-1 +-2 +-2 +-1 +0 +1 +2 +1The feasibility problem is formalized in Equation 11, as follows: +min +0 , +(11a) +s.t. +o +� +i=1 +λivi = vk , +(11b) +o +� +i=1 +λi = 1 , +(11c) +λk = 0 , +(11d) +λi ≥ 0 , ∀i ∈ {1, . . . , o}, +(11e) +where, for each vertex candidate, denoted by vk, we search for a λ vector such that vk can be +described as a convex combination of the remaining vertices. If that is the case, then vk is not +a vertex and it can be removed. Otherwise, the point is a vertex and must remain in the set of +vertices. The non-vertex elimination is formalized in Algorithm 5. +Algorithm 5: Removing Internal Points (RP) +Input: V ∈ Ro×n, V = [v[1], . . . , v[o]], v[i] ∈ Rn , ∀i ∈ {1, . . . , o} +Function RP(V): +for k ∈ {1, . . . , o} do +if ¬∃λ ∈ Ro : �o +i=1 λiv[i] = v[k] ∧ �o +i=1 λi = 1 ∧ λk = 0 ∧ λi ≥ 0 , ∀i ∈ {1, . . . , o} +then +V ← [v[1], . . . , v[k − 1], v[k + 1], . . . , v[o]] +end +return V +The output of the process of removing non-vertices can be viewed in Figure 7. As we can see, +instead of returning the exact non-convex set, this algorithm computes an over-approximation for +the output set which is simpler than the exact output as it is a unique set. +Approximate Polytope Network Mapping +Finally, we show the complete process, presented by the Algorithm 6, to compute an approxima- +tion for the output reachable set for a given input polyhedron P. +3.3 +Exact polytope network mapping (EPNM) +We present in this section the second proposed approach, which similarly to the first one computes +the reachable set by using the vertices of a given input polytope, but computes the exact output +instead. We consider the same input closed convex polytope P and the same set of vertices V as +presented in the previous section. This approach comprises six parts: +1. Affine map of the vertices with weights and biases; +2. Adjacent vertex identification; +3. Origin verification; +11 + +Figure 7: Representation of the convex hull of the output of the ReLU mapping. +Algorithm 6: Simple Approximated Politope Network Mapping +Input: V ∈ Ro×n, Wl and θl for l ∈ {1, . . . , L} +Function SAPNM(V, W, θ): +Z[0] ← V +for l ∈ {1, . . . , L} do +ˆZ[l] ← AM(Z[l − 1], Wl, θl) /* Compute affine mapping for layer l +*/ +if l < L then +E[l] ← EI(ˆZ[l]) /* Compute edge-skeleton of polyhedron given by +vertices ˆZ[l] +*/ +ˆZ[l] ← II(ˆZ[l], E[l]) /* Obtain intercept of edges with horthant +hyperplanes +*/ +ˆZ[l] ← ReLU(ˆZ[l]) /* Apply ReLU mapping to the vertex set ˆZ[l] +*/ +Z[l] ← RP(ˆZ[l]) /* Remove non-vertex points from vertex set Z[l] +*/ +end +return Z[L] +4. Polytope intersection with orthant’s hyperplanes; +5. Separate points according to the orthant to which they belong; +6. ReLU mapping; and +7. Removing non-vertices; +In this section we only present parts 3 and 5, which were not previously introduced. +Origin verification +After we compute the edges of P, we need to verify if the origin is inside it. In case it is true, then +we need to insert it into the set of vertices that will be partitioned with respect to the orthant that +12 + +2 +V +1 +VA +0 +V2 +-1 +-2 +-2 +-1 +0 +1 +2 +1they belong in the next part of the algorithm. This verification is compulsory due to the separation +of vertices according to the orthant to which they belong (for instance, if the origin is in P and is +not in the set of the vertices of the partitions of the polytope, then the union of the partitions will +not be equal P), as presented in Figure 8. Notice that, each partition represent the portion of P +that is inside an orthant, i.e., the portion of P that is in the positive orthant, in Figure 8, is the +union of the red area and the shaded area. We can see that if the origin is not part of the set of +vertices of each partition of P, after the orthant separation process, the shaded area will be missing +for the portion of P inside the positive orthant. Thus, we need to include the origin point in the +respective set so that the portion of P in the positive orthant also includes the shaded area. To +Figure 8: Representation of the issue that will occur if the origin is not inserted to the vertex set +for the orthant separation process. +perform such a verification, we state the search problem as a feasibility analysis, in which we try to +verify if there exists a vector λ ∈ Ro such that the origin is a convex combination of the vertices in +V. This problem is formalized by Equation (12). +min +0 , +(12a) +s.t. +o +� +i=1 +λivi = 0 , +(12b) +o +� +i=1 +λi = 1 , +(12c) +λi ≥ 0 , ∀i ∈ {1, . . . , o}. +(12d) +Algorithm 7 formalizes the search process associated with the verification of the origin. +13 + +V +2 +V +V +3 +X +V +6 +V +5Algorithm 7: Origin Search (OS) +Input: V ∈ Ro×n, V = [v[1], . . . , v[o]], v[i] ∈ Rn , ∀i ∈ {1, . . . , o} +Function OS(V): +if ∃λ ∈ Ro : �o +i=1 λivi = 0 ∧ �o +i=1 λi = 1 ∧ λi ≥ 0 , ∀i ∈ {1, . . . , o} then +V ← [v[1], . . . , v[o] | 0] /* Adding the origin to the vertex set +*/ +return V +Separating points to their respective orthants +We present in this section the process of splitting the vertex set V into several sets, each associated +with a different orthant. In other words, we split V in such a way that each resulting set contains +points located in a single orthant (observe that the convex hull of each set of vertices represents the +partition of P that is inside an orthant). We begin by computing: +b = 1 +2(sign(vi) + 1|vi|) +(13) +where, vi is the vertex under analysis, sign : Rn → Rn is the sign mapping previously presented +and 1|vi| is a vector with all the components equal to one and has size |vi|. +The elements of b with value equal to 1 are associated with a positive element of vi, the elements +equal to 0 are associated with negative components of the vertex, and those components of b equal +to 1/2 are associated with the null components of the vertex. Notice that a null component of a +vertex means that this point belongs to at least two different orthants (the origin being a special +case inside all the orthants). +For those cases where the component of b is equal to 1/2, we must guarantee that the associated +vertex vi is properly inserted into each of those sets that are associated with the orthants it belongs +to (i.e., if vi has one null component, it must be inserted in two sets). Then, the algorithm starts a +verification process to identify those component of b that are equal 1/2. In case it is equal true, two +copies of vi must be created. This verification process is repeated until all components of b have +been checked. This process is detailed in Algorithm 8. +After checking for null values in vi, the sets in which it must be placed must be defined. It should +be noted that at the end of this process, b is a binary vector that represents the index in which +vi must be placed (there might be more than one b for a single vertex). Equation (14) presents +the process for converting b into a decimal number q. The calculation of the index is presented in +Algorithm 9. +q = +n +� +i=1 +bi2i−1 +(14) +Figure 9 illustrates a toy example of how this process works for a single vertex vi. +In this +example, vi = (−2, 0), with the second component being null. As a result, b = (0, 1 +2). Next, two +copies of b are created. The components of the first (second) copy that are equal to 1 +2 are replaced +by 0 (1). In this case, as there was only one null component, the generated vectors are (0, 0) and +(0, 1). The vector generated from b contains the binary representation of the indices of the sets +that vi must be placed in. In this example, vi must be placed in the orthants with indices 0 and +2. It is important to note that the indices used by the algorithm to identify the destination sets are +different from the traditional orthant enumeration (2nd and 3rd quadrants in this case). +The orthant separation complete process is computed accordingly by Algorithm 10. +Continuing the example from the previous subsection, the orthant separation performed by +Algorithm 10 will divide the polytope illustrated in Figure 5 into four separate polytopes, one +14 + +Algorithm 8: Zeros Verification (ZV) +Input: b ∈ [0, 1]n +Function ZV(b): +A ← {b} +B ← {b} +while |B| ̸= 0 do +B ← ∅ +for a ∈ A do +for i ∈ {1, . . . , |a|} do +/* Check for those components that are equal 1/2 +*/ +if a[i] = 1/2 then +b ← a +b[i] ← 1 +B ← B ∪ b /* Insert the first copy into B +*/ +b ← a +b[i] ← 0 +B ← B ∪ b /* Insert the second copy into B +*/ +break +end +end +if |B| ̸= 0 then +A ← B +end +return A /* return a set of vectors +*/ +Algorithm 9: Get Array Position (AP) +Input: b ∈ {0, 1}m +Function AP(b): +let q be an integer +q ← 0 +for i ∈ {1, . . . , n} do +q ← q + b[i]2i−1 +end +return q +for each orthant, as shown in Figure 10. It is worth noting that vertices v5, v6, v7, v8, and v9 have +been placed in more than one set, as they are located in more than one orthant. +Exact Polytope Network Mapping +The layer mapping and the complete process for the Exact Polytope Network Mapping is formal- +ized in Algorithm 11. The algorithm takes as input the vertices of the input polytope aimed to be +verified. Then, it computes the affine map of these vertices (calculated by the algorithm AM) and +identifies the edges between adjacent vertices (with the algorithm EI). For the vertices connected +by an edge, the algorithm computes the intersection of the corresponding edge with each orthant’s +supporting hyperplane, when those vertices are not in the same orthant (by means of the algorithm +15 + +Figure 9: Vertex separation process for a single vertex. Given a vertex vi = (−2, 0), the associated +b is given by (0, 1 +2). As we have that there is a component of b equal to 1 +2, we split it into two +vectors in such a way that the component with value equal to 1 +2 is substituted by 1 and 0 in the +newly created vectors. These new vertices contain the binary representation of the orthant’s index +that vi belong to. In the presented example, vi belong to 2 different orthants (the blue and the +green ones). +Algorithm 10: Separate points per orthant (SP) +Input: V ∈ Ro×n, V = [v[1], . . . , v[o]], v[i] ∈ Rn , ∀i ∈ {1, . . . , o} +Function SP(V): +let Z[1 . . . 2n] be an array +for i ∈ {1, . . . , n} do +let b[1, . . . , n] be an array +b ← 1 +2(sign(v[i]) + 1|v[i]|) +% identify if there are null coordinates +B ← ZV(b) /* Identify the orthants to which v[i] belongs +*/ +for b′ ∈ B do +q ← AP(b′) /* Calculate index position of orthant b′ to which v[i] +belongs +*/ +if Z[q] = ∅ then +Z[q] ← v[i] +else +Z[q] ← [Z[q] | v[i]] /* Appending the new vertex +*/ +end +end +return Z +II), and verifies whether or not the origin belongs to the polytope under analysis (checked by the +algorithm OS). Next, it separates all of these points in different sets, where each set contains those +vertices that belong to a single orthant (computed by the algorithm SP). Finally, the algorithm per- +forms the ReLU mapping and removes non-vertices for each partition generated. As later presented, +16 + +X2 +v, = (-2,0) +b = (0,/2) +V, = (-2,0) +(0,0) +(0,1) +X1 +q = 0*21+0*2° = 0 +q = 1*21+0*2° = 2 +(3rd orthant) +(2nd orthant)Figure 10: Representation of the separation of the polytope P after the application of the affine +map. +the ReLU mapping preserves the convexity inside a given orthant, though there are points that are +not vertices after the application of the non-linear mapping (check vertex v3 in Figure 6). +Finally, after the application of the ReLU mapping and the removal of non-vertices (RP) for +each partition in the set, as illustrated in Figure 10, the result of the EPNM algorithm for a single +layer mapping is shown in Figure 11. Observe that this output, which is a union of sets, represents +the mapping of a unique layer. For each of the sets that result from the previous layer, the algorithm +computes the exact mapping for the next layer. This process will repeat until the algorithm maps +all the sets to the last layer of the neural network. For instance, considering that the visual problem +has one extra layer, each of the four output sets (one set for each orthant, as the intersection of +P with each orthant is not empty) will be mapped to the next layer following the same described +process. +3.4 +Partially approximated polytope network mapping (PAPNM) +We have proposed a third approach, which involves a slight modification of the EPNM algorithm. +Instead of an exact mapping between layers, this approach allows for the merging of some of the +resulting sets. By computing the convex hull of the union for some of the output sets, we aim to +achieve a more accurate approximation of the output set (compared to APNM) while also reducing +the execution time of the algorithm (compared to EPNM). The result is presented in Algorithm 12. +The main difference between Algorithm 11 and Algorithm 12 is the application of the MS +procedure after the separation process. The MS algorithm comprises the merging procedure, where +a set of sets of vertices is given as input. As each set of vertices represents a single polytope, this +procedure merges some of these sets of vertices to produce fewer sets compared to the exact mapping. +It is worth noting that the output of MS is an overapproximation of the input set it receives, and +that the extreme case in which a single set of vertices is computed is exactly the case of APNM. +Here, we propose a basic approach for the merging process in Algorithm 13. The MS algorithm +takes the set P which contains the sets of vertices and an integer d as inputs. The sets of vertices +are then grouped into sets of size d and each group is merged (i.e, for the case where P has 13 sets +17 + +2 +V1 +1 +VA +V +0 +NS +-1 +V6 +V3 +-2 +-2 +-1 +0 +1 +2 +1Algorithm 11: Exact Polytope Network Mapping +Input: V ∈ Ro×n, /* vertices of the polytope +*/ +Wl and θl for l ∈ {1, . . . , L} /* network layers’ weights and biases +*/ +Function EPNM(V, W, θ): +let P be an empty set +P ← {V} /* The list of polytopes’s vertices; starting with the input +polytope +*/ +for l ∈ {1, . . . , L} do +ˆP ← ∅ /* ˆP is the auxiliary set that represents P in the next layer */ +for Z ∈ P do +/* iterate for each set Z of vertices in P +*/ +if l > 1 then +Z ← ReLU(Z) /* Apply ReLU mapping to the vertex set Z +*/ +Z ← RP(Z) /* Remove non-vertex points from vertex set Z +*/ +Z ← AM(Z, Wl, θl) /* Perform affine mapping, Z ← WlZ + θl +*/ +if l < L then +E ← EI(Z) /* Compute edge-skeleton E of the polyhedron given by +vertices Z +*/ +Z ← II(Z, E) /* Add edge intercepts with orthant’s supporting +hyperplanes to vertex set +*/ +Z ← OS(Z) /* Add origin to the vertex set if needed +*/ +Z ← SP(Z) /* Transform the vertex set into a list of vertices +for each orthant, considering the orthants to which each +vertex belongs +*/ +for k ∈ {1, . . . , |Z|} do +if Z[k] ̸= ∅ then +ˆP ← ˆP ∪ Z[k] +end +end +P ← ˆP +end +return P +and d = 3, there will be 4 groups of 3 sets and 1 group of 1 set). It is important to note that this +is just a simple example of the merging procedure and that different heuristics can be implemented +to improve this process. +The use of this merging procedure allows for different levels of approximation, resulting in a +range of possible approximations from the exact mapping (EPNM) to the coarser case (APNM). +It is important to ensure that the merging procedure returns an overapproximation of the exact +mapping of each layer, which is a necessary condition to ensure the soundness of PAPNM. +4 +Demonstrations +We present in this section demonstrations that provide theoretical guarantees for the correctness +of each proposed algorithm. +18 + +Algorithm 12: Partially approximate polytope network mapping +Input: V ∈ Ro×n, /* vertices of the polytope +*/ +Wl and θl for l ∈ {1, . . . , L} /* network layers’ weights and biases +*/ +/* d is the size of each group of sets that will be merged +*/ +Function PAPNM(V, W, θ, d): +let P be an empty set +P ← {V} +for l ∈ {1, . . . , L} do +ˆP ← ∅ +for Z ∈ P do +if l > 1 then +Z ← ReLU(Z) +Z ← RP(Z) +Z ← AM(Z, Wl, θl) +if l < L then +E ← EI(Z) +Z ← II(Z, E) +Z ← OS(Z) +Z ← SP(Z) +Z ← MS(Z, d) /* new function for merging sets +*/ +for k ∈ {1, . . . , |Z|} do +ˆP ← ˆP ∪ Zk +end +end +P ← ˆP +end +return P +19 + +Figure 11: Representation of the output reachable set for a single layer of the EPNM algorithm. +Algorithm 13: Merge Sets (MS) +Input: P = {V1, . . . , Vq}, Vi ∈ Roi×n, i ∈ {1, . . . , q} /* set of polytopes’ vertices */ +d ∈ N /* d is the size of each group of sets that will be merged +*/ +Function MS(P, d): +let A be an empty set +for i ∈ {1, . . . , +� +|P| +d +� +} do +let B[1 . . . n] be an array +for j ∈ {(i − 1) × d + 1, . . . , min(i × d, |P|)} do +if B = ∅ then +B ← V[j] +else +B ← (B|V[j]) +end +A ← A ∪ B +end +return A +4.1 +Identification of adjacent vertices +Let P = {�o +i=1 λivi | �o +i=1 λi = 1, λi ≥ 0 e vi ∈ V, ∀i ∈ {1, . . . , o}} be a convex V-polytope de- +fined as the convex combination of its vertices, where V = {v1, . . . , vo} is the set of vertices of a +polyhedron P. +Definition 1. Given a vector c ∈ Rn and δ = max{cT x | x ∈ P}, we have that H = {x | cT x = δ} +is a supporting hyperplane of P. +Definition 2. F is a face of P if F = P or F = P ∩ H for some supporting hyperplane H. In other +words, F is a face of P if, and only if, F is the set of optimal solutions for max{cT x | x ∈ P} for a +given c ∈ Rn [21]. +Definition 3. vi e vj are adjacent vertices if there is a vector c ∈ Rn such that cT vi = cT vj = +20 + +2 +V +1 +V4 +V9 +0 +V5 +-1 +-2 +-2 +-1 +0 +1 +2 +1max{cT x | x ∈ P} > cT vk, ∀k ∈ {1, . . . , o}, k ̸= i and k ̸= j [21]. +Notice that the face F is an edge in the particular case stated by Definition 3. We propose that: +Proposition 1. Two extreme point vi, vj ∈ V are adjacent if, and only if, it is not possible to +compute the median point, ¯v = (vi + vj)/2, as a convex combination of the vertices in V \ {vi} and +V \ {vj}. +Proof. Given two vertices vi and vj, we have two possibilities regarding their adjacency: 1) they +are adjacent, or else 2) they are not adjacent. +For the first case, as vi and vj are adjacent, we have by definition that there is a hyperplane +Ha = {x | cT x = d} that contains both vi and vj, such that Ha ∩ P = max{cT x | x ∈ P} > +cT vk, ∀k ∈ {1, . . . , o}, k ̸= i and k ̸= j for a given c ∈ Rn. Therefore, all the remaining extreme +points from P, except from vi and vj, are in the open half-space Hb = {x | cT x < d}. Consequently, +the median point of vi and vj, denoted by ¯v = (vi + vj)/2, can not be expressed as a convex +combination of vk, for k ∈ {1, . . . , o} \ {i} or k ∈ {1, . . . , o} \ {j}. +Considering that vi and vj are not adjacent, let Pi and Pj be two polytopes given by the convex +combination of the extreme points V \ {vi} and V \ {vj}, respectively. Hence, there are two distinct +possibilities: ¯v ∈ Pj or ¯v ̸∈ Pj. For the first case, as ¯v ∈ Pj, then we can compute this point as +a convex combination of the vertices in V \ {vj}. For the second case, as ¯v ̸∈ Pj and ¯v ∈ P, then +¯v ∈ P \ Pj. However, since the vertices that are adjacent to vj are in V \ {vi, vj}, then P \ Pj ⊆ Pi. +From this fact it follows that ¯v ∈ Pi and, consequently, ¯v can be expressed as a convex combination +of the vertices V \ {vi}. +4.2 +ReLU convexity inside an orthant +Let f : Rn → Rn be a function that denotes the ReLU mapping, defined by the Equation (15): +f(x)i = max(0, xi) +(15) +Proposition 2. Given x, y ∈ Rn, if sup{γ | γ = sign(x)i − sign(y)i, ∀i ∈ {1, . . . , n}} ≤ 1, or, in +other words, if x and y belong to the same orthant, we have that f(x + y) = f(x) + f(y) and that +f(αx) = αf(x). +Proof. Denoting each orthant of Rn as Oj, ∀j ∈ {1, . . . , 2n}, we can rewrite the ReLU mapping, +given by f, as fj : Oj → Rn, such that: +fj(x) = Λjx +(16) +where Λj ∈ Rn×n is the matrix in which the elements of the principal diagonal associated with a +negative component of x ∈ Oj are equal 0, while the remaining elements are equal 1 (λk,l for k = l). +The elements off the principal diagonal are all equal to zero (λk,l = 0 for all k ̸= l). +Therefore, fj preserves convexity since it consists of a linear mapping that satisfies both properties +stated in the proposition, the addition property: +fj(x + y) = Λj(x + y) += Λjx + Λjy += fj(x) + fj(y) +21 + +and the product by a scalar: +fj(αx) = Λj(αx) += αΛjx += αfj(x) +where α ∈ R. +Hence, as fj satisfies the linearity conditions within each orthant Oj and the linear mapping +preserves convexity, it follows that: +fj(θx + (1 − θ)y) = Λj(θx + (1 − θ)y) += θΛjx + (1 − θ)Λjy += θfj(x) + (1 − θ)fj(y) +for all x, y ∈ Oj e θ ∈ [0, 1]. Therefore, the ReLU mapping preserves the convexity inside a given +orthant. +4.3 +V-polytope and half-space intersection +Let P = {�o +i=1 λivi | �o +i=1 λi = 1, λi ≥ 0 and vi ∈ V, ∀i ∈ {1, . . . , o}} be a closed convex poly- +hedron defined in terms of the convex combination of its vertices (or extreme points), where +V = {v1, . . . , vn} is the set of the vertices of P. +Such a polyhedron can also be defined as a +set of inequalities P = {x | Cx ≤ d}, such that C ∈ Rm×n and d ∈ Rm. +(a) P ∩ H = P +(b) P ∩ H ⊂ P +(c) P ∩ H = ∅ +Figure 12: Representation of all the three different possible cases with respect to the intersection of +P with a half-space H. +There are three different possible cases regarding the intersection of P with a half-space H = +{x | aT x ≤ b}, where a ∈ Rn and b ∈ R (Figure 12 presents a visual representation for each case): +1. P ∩ H = P; +2. P ∩ H ⊂ P; +3. P ∩ H = ∅. +The first and the third cases are trivial. For the first case, we have that all the extreme points +of P are in H. For the third case none of the extreme points of P belongs to H. +22 + +H +2 +1 +V +3 +P +v +4 +v +6 +V +5H +2 +3 +P +4 +V +6 +v +5H +2 +3 +p +V +7 +4 +v +6 +5For the second case, a new face is generated for P, given by Ps = P ∩ Hs, where Hs = {x | +aT x = b} such that a ∈ Rn and b ∈ R. Note that Hs is the supporting hyperplane of H. Therefore, +those vertices of P that are not in H, are not extreme points of P ∩ H. Thus, it is necessary to find +those extreme points of the new polytope Ph = P ∩ H. Figure 13 presents a visualization of the +elements previously defined. +Figure 13: Representation of the elements of interest from the intersection between P and H for +the second case. The supporting hyperplane Hs of H is presented in blue. In green we have the +intersection between Hs and P, and in red the result of the intersection between P with H. Notice +that, for this example, Vh = {v1, v2, v3, v4}. +We denote by Vh the subset of vertices of P that belong to H, given by Vh = V ∩ H. +Let +E = {E1, . . . , Ep} be the set of edges of P. Ei denotes the set of points that belong to the i-th edge +of P. Finally, we denote by Vp the set of vertices from Ph. Observe that Vh ⊆ Vp. +Let c be a non-zero vector and δ = max{cT x | Cx ≤ d}. The affine hyperplane Ha = {x | +cT x = δ} is a supporting hyperplane of P. +Definition 4. A subset F of P is called a face of P if F = P or F = P ∩ Ha [22]. +Definition 5. F is a face of P if, and only if, F is not empty and F = {x ∈ P | C′x = d′}, for a +subsystem C′x ≤ d′ of Cx ≤ d [22]. +Proposition 3. Considering that P ∩ H ⊂ P and given that Vp is the set of extreme points of Ph, +if v ∈ Vp and v ̸∈ Vh then v ∈ Ps. +Proof. By definition, an extreme point of a polytope is a 0-dimensional face, thus: +F = {x | C′x = d′} +(17) +where C′x ≤ d′ is a subsystem of Cx ≤ d. Remember that P = {x | Cx ≤ d} and that H = {x | +aT x ≤ b}. Furthermore, since a vertex v is the particular case of a face given by the intersection of +n hyperplanes, for a n-dimensional space, then: +v = {x | C′′x = d′′} +(18) +for some C′′ ∈ Rn×n and d′′ ∈ Rn, such that det(C′′) ̸= 0 and v ̸= ∅. +However, for x ∈ H \ Hs, no new solution for the subsystem C′′x = d′′ subsystem of: +� C +aT +� +x ≤ +�d +b +� +(19) +23 + +H +2 +H +P +V. +S +3 +P +P +4 +V +6 +v +5such that det(C′′) ̸= 0 and v ̸= ∅, is possible. Put in other words, as no constraint was placed in +H \ Hs, no new vertex was generated in P ∩ (H \ Hs). Consequently, those vertices generated by +the intersection with H, in case they exist, must belong to Hs. Hence, if v ∈ Vp and v ̸∈ Vh, then +v ∈ Ps. +Proposition 4. Considering that P ∩ H ⊂ P, if Vp is the set of extreme points of Ph, then +Vp \ Vh = E ∩ Hs. +Proof. As P ∩ H ⊂ P, then Ps is a face of Ph. Furthermore, if F is a face of Ps, then F is a face +of Ph. Therefore, the 0-dimensional faces of Ps (vertices) are also faces of Ph. +Recall that a vertex is defined by the intersection of n hyperplanes, for a n-dimensional space, +that P = {x | Cx ≤ d} and that H = {x | aT x ≤ b}. Therewith, there are two possible cases for +the vertices of Ps: +• v is given by C′x = d′, where C′x ≤ d′ is a subsystem of Cx ≤ d; +• v is given by C′′x = d′′, where: +C′′ = +�C′ +aT +� +, d′′ = +�d′ +b +� +(20) +For the first case, we trivially observe that v ∈ Vh. For the second case, as C′′ ∈ Rn×n then +C′x = d′ represents the intersection of n − 1 hyperplanes, resulting in an edge of P, given by +Ei = {x | C′x = d′}. Hence, for the case where P ∩ H ⊂ P, if Vp is the set of vertices of Ph, then +Vp \ Vh = E ∩ Hs. +4.4 +Origin regarding polytope intersection +As presented in the previous section, considering that P is a closed polytope defined by the convex +combination of its vertices, the intersection between P and a half-space H is given by the convex +combination of the vertices from P in the intersection with H, along with the vertices obtained by +the intersection of the supporting hyperplane of H with the edges of P. +Consequently, the intersection between P and Oj, where Oj represents one of the 2n orthants +in a n-dimensional space, can be rewrite as P ∩ H1 ∩ · · · ∩ Hn. Note that Oj = H1 ∩ · · · ∩ Hn for +suitable half-spaces H1, . . . , Hn. By induction, it can be shown that the vertices of P ∩ Oj consist +of the union of vertices of P that also belong to Oj with the vertices given by the intersection of the +edges of P with the supporting hyperplanes of Oj (denoted as Hi, ∀i ∈ {1, . . . , n}). We define Oj +as: +Oj = {x | Φjx ≤ 0} +(21) +where Φj is the suitable matrix given by: +Φj = +� +���� +φj,1,1 +0 +. . . +0 +0 +φj,2,2 +. . . +0 +... +... +... +... +0 +0 +. . . +φj,n,n +� +���� +(22) +such that φj,i,i ∈ {−1, 1}, ∀i ∈ {1, . . . , n}. +We can see that the equation system Φjx = 0 has the origin as its sole solution for any orthant +j. Therefore, the unique extreme point of Oj is the origin for all j ∈ {1, . . . , 2n}. +24 + +Proposition 5. If the origin belongs to P, then it is an extreme point of P ∩ Oj, for all j ∈ +{1, . . . , 2n}. +Proof. Let P be a convex closed polytope and Oj a convex cone. It is known that the intersection +between P and Oj is a closed convex polytope. We have that the vertices from P that belong to Oj +are also vertices of P ∩ Oj, denoted by VOj. Also, we have that P ∩ Oj = P ∩ H1 ∩ · · · ∩ Hn. +Based on Proposition 4, we have by induction that the intersection of the edges of P with the +supporting hyperplanes of Oj are also vertices of P ∩ Oj, denoted by Vh. +For the origin, which is the single vertex of Oj, there are two possible cases: +1. the origin is not in P: this is the trivial case in which the origin can not be a vertex of P ∩ Oj, +as it is not in P; +2. the origin is in P: in this case, by contradiction we suppose that the origin is not a vertex of +P ∩ Oj, denoted by 0. Then, there must exist two points v, u ∈ P ∩ Oj, such that: +0 = λv + (1 − λ)u +(23) +given that 0 < λ < 1, since 0 ̸= v and 0 ̸= u. As the origin is given by 0 = (0, 0, . . . , 0, 0), +there must exist a solution for the equation 0 = λvi + (1 − λ)ui for all i ∈ {1, . . . , n}, such +that: +vi = (λ − 1)ui +λ +(24) +Since (λ−1) +λ +< 0, the sign of vi and ui must by different if vi ̸= 0 and ui ̸= 0. However, inside a +given orthant there must not exist a point with components that have opposite signs. Hence, +by contradiction, if the origin is in P, it must be a vertex of P ∩ Oj. +4.5 +Correctness of APNM algorithm +Let F : Rn → Rm denote a neural network mapping, where L is the number of layers, the +weights of the layer l are given by Wl ∈ R|xl|×|xl−1| and the biases by θl ∈ R|xl|. +Thereby, +F(x) = (FL ◦ FL−1 ◦ · · · ◦ F1)(x) where Fl(x) = ReLU(Wlx + θl) is the mapping for each layer +l ∈ {1, . . . , L − 1}. The only difference for the last layer is the activation (usually sigmoid for binary +problems, or softmax for multi-class problems). +For a given layer l, we have that the set Xl denotes the inputs that we aim to map regarding Fl, +such that Xl = {�o +i=1 λivi | �o +i=1 λi ∧ λi ≥ 0 ∧ vi ∈ Vl, ∀i ∈ {1, . . . , o}} and Vl is the set of vertices +of Xl. Associated with Xl, there is Yl = {Fl(xl) | xl ∈ Xl}, which corresponds to the output set for +layer l. +Finally, let �Fl denote the output mapping for a given layer by the Algorithm 6, given by: +�Fl(Vl, Wl, θl) = RP +� +ReLU +� +II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))) +�� +(25) +where AM corresponds the affine map given by Algorithm 1, EI is the edge identification given +by the Algorithm 2, II is the intersection identification defined by Algorithm 3, ReLU is given +by Algorithm 4, and RP stands for removing non-vertices defined by Algorithm 5. We denote the +convex hull mapping for a given set of vertices by CH. +25 + +Proposition 6. Given a closed convex polytope Xl as input set and Vl as the set of its vertices, +then it implies that Yl ⊆ CH( �Fl(Vl, Wl, θl)). In other words, every output of a layer l associated +with an input in Xl is in the convex hull of �Fl(Vl, Wl, θl) +Proof. The mapping of a layer l from F is composed of an affine map (Wlxl + θl) and a non-linear +map (ReLU). Hence, since Xl is a closed convex polytope and Vl its vertices, the affine map of Xl +is given by the convex hull of the affine map of its vertices, as computed by Algorithm 1. +As previously presented, the ReLU map is non-linear and therefore does not necessarily preserve +the convexity of a given input set. However, as established by Proposition 2, the ReLU mapping +preserves the convexity of a convex set inside a given orthant. +Therefore, we divide the resulting set of the affine map in such a way that each partition is inside +a single orthant, so that we can apply the ReLU map to the set � +Xl = CH(AM(V, Wl, θl)). As +assured by Proposition 4, the intersection of a given orthant Oj with the polytope � +Xl consists of the +convex hull of the union of the vertices of � +Xl that are in Oj, with the vertices from the intersection +of the edges of � +Xl with the supporting hyperplanes that define Oj. Note that Oj represents a given +orthant, for all j ∈ {1, . . . , 2|xl|}. +Thus, we first need to compute the edges of � +Xl, calculated by EI(AM(V, Wl, θl)), as stated by +Proposition 1, followed by the determination of the intersection of these edges with the supporting +hyperplanes of Oj, given by II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))) and computed by Algorithm +3. +Therewith, we have that � +Xl was divided in such a way that each partition is inside a single +orthant. Then, the ReLU mapping is applied to the vertices of each partition of � +Xl, resulting in +ReLU(II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl)))). Proposition 2 assures that the ReLU mapping +preserves the convexity inside a single orthant, which allows its previous application. Note that all +the vertices will be in the non-negative orthant after applying the ReLU mapping. +Finally, we compute the convex hull of all the output sets of the ReLU mapping (at most 2|xl|) by +removing those points that are not vertices of CH(ReLU(II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))))). +This final step is computed by Algorithm 5 (RP) as stated in Equation (25). +Consequently, CH( �Fl(Vl, Wl, θl)) is the convex hull of the exact map Yl of Xl. +We denote by �F the composition of �Fl for all layers of the neural network, except the last one, +where the non-linear map is not applied, given by �F(V, W, θ) = ( �FL ◦ �FL−1 ◦ · · · ◦ �F1)(V, W, θ). +Furthermore, we define Y = {F(x) | x ∈ X} as the exact output set of the network, regarding the +closed convex input polytope X and its corresponding set of vertices V. +Proposition 7. Given a closed convex polytope X as the input set and V, the set of its vertices, then +we have that Y ⊆ CH( �F(V, W, θ)). Put in different terms, each output of the network associated +with an input in X is in �F(V, W, θ). +Proof. As established by Proposition 6, Yl ⊆ CH( �Fl(Vl, Wl, θl)) for a given layer l of the neural +network F. Then, for l = 1, it follows that: +Y1 ⊆ CH( �F1(V1, W1, θ1)) +(26) +where V1 is the set of vertices from X1 and X1 = X. As the output of the first layer is the input of +the second one, we have that X2 = CH( �F1(V1, W1, θ1)) and that V2 = �F1(V1, W1, θ1). +For l = 2: +Y2 ⊆ CH( �F2(V2, W2, θ2)) +(27) +26 + +Then, by replacing V2 with the output of layer 1, results in: +Y2 ⊆ CH( �F2( �F1(V1, W1, θ1), W2, θ2)) +(28) +Now, for layers k and k + 1, it follows by induction that: +Yk+1 ⊆ CH( �Fk+1( �Fk(Vk, Wk, θk), Wk+1, θk+1)) +(29) +Consequently, the mapping ˆF in fact computes an over-approximation for the exact output set +Y. +4.6 +Correctness of EPNM algorithm +The set Yl denotes the exact output associated with the input set Xl, regarding the layer l of +the neural network F. The mapping of a given layer l implemented by Algorithm 11, denoted here +by �El, is stated as: +�El,k(Vl, Wl, θl) = ReLU +� +SP +� +OS +� +II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))) +�� +k +� +(30) +where AM is the affine map computed by Algorithm 1, EI is the edge identification implemented +by Algorithm 2, II is the intersection identification given by Algorithm 3, ReLU is computed by +Algorithm 4, OS is implemented by Algorithm 7 to verify if the origin is in the polytope, and +finally SP, which separates the vertices in orthants, is computed by Algorithm 10. Notice that +k ∈ {1, . . . , K} is the index that represents each �El(Vl, Wl, θl). We denote the convex hull mapping +for a given input set of vertices by CH. +Proposition 8. Given a convex closed polytope Xl as input set and Vl, the set of its vertices, we have +that Yl = �K +k=1 CH( �El,k(Vl, Wl, θl)), where K is the number of sets that comprise �El(Vl, Wl, θl). +Put another way, the set of outputs of the layer l, resulting from all inputs in Xl, consists of the +union of the convex hull of each set �El,k(Vl, Wl, θl). +Proof. As presented previously, the mapping of a given layer l of the neural network F is a com- +position of two different functions: one affine mapping (Wlxl + θl) with one non-linear mapping +(ReLU). As Xl is a closed convex polytope, the affine mapping is obtained by the convex hull of the +affine map of each of its vertices, implemented by Algorithm 1. +ReLU does not necessarily preserve the convexity of a given input set. However, as shown by +Proposition 2, the ReLU mapping preserves the convexity of an input set if the input is inside a +single orthant. +Note that the affine mapping of the input set is computed by AM(Vl, Wl, θl), where � +Xl = +CH(AM(Vl, Wl, θl)) denotes the application of the affine mapping in Xl. Therefore, it is necessary +to split � +Xl in such a way that each partition lies inside a single orthant, so that it becomes possible +to apply the ReLU mapping to the vertices of � +Xl. As assured by Proposition 4, the intersection of +an orthant Oj with a polytope � +Xl consists of the union of the vertices of � +Xl that are in Oj, with the +vertices in the intersection of the edges of � +Xl with the supporting hyperplanes of Oj and the origin, +if the latter lies inside � +Xl, according to Proposition 5. +Thus, we firstly compute the edges of � +Xl, a step denoted by EI(AM(Vl, Wl, θl)) and computed +by Algorithm 2, as stated by Proposition 1. Then, the intersection of these edges with the supporting +hyperplanes of orthant Oj is obtained with Algorithm 3, and finally Algorithm 7 verifies whether +the origin belongs to � +Xl. The result of such an operation is given by: +OS +� +II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))) +� +(31) +27 + +In the next step, we separate the vertices of � +Xl in different sets, such that those vertices in the +same set represent the portion of � +Xl that is inside a single orthant. This process takes place to +enable the application of the ReLU mapping, as the separation allows the non-linear mapping to be +applied in each partition of � +Xl while ensuring convexity, as established by Proposition 2. Algorithm +10 performs the partitioning operation, given by: +SP +� +OS +� +II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))) +�� +(32) +The result of SP is the set of sets of vertices, where the convex hull of each set represents a +partition of � +Xl restricted to a single orthant. Thus, �El(Vl, Wl, θl) denotes the result of the application +of the ReLU mapping to each of these sets. Finally, we have that �El(Vl, Wl, θl) represents the +exact mapping of Xl, as we applied both the affine and the non-linear mapping without any over- +approximation. Consequently, Algorithm 11 in fact returns the exact output set for a given layer l +of the neural network F. +Finally, we denote by �E the composition of �El, for each l ∈ {1, . . . , L}. To avoid repetition, the +Proposition 9 is not presented. However, it follows the same inductive process as presented for the +APNM (Proposition 7). +Proposition 9. Given a closed convex polytope X as input set and V, the set of its vertices, we +have that Y = �K +k=1 CH( �Ek(V, W, θ)), where K is the number of sets in �E(V, W, θ). In other +terms, the set of outputs of the neural network F, associated with the input set X, is in the union +of the convex hull of each set �Ek(V, W, θ). +5 +Application +In this section we present a comparative analysis between the proposed vertex-based reachability +approach and representative algorithms from the literature. +This comparison is carried out by +verifying one of the properties from ACAS XU [1]. +5.1 +ACAS XU +The Aircraft Collision Avoidance System (ACAS XU) [1] comprises a set of fully connected +neural networks that aim to eliminate the possibility of collisions between two aircraft. The system +comprises 45 trained models, where each model receives as inputs five properties of the ownship +and the intruder (the aircraft that is invading the space of the ownship): the distance from the +ownship to the intruder (ρ), the angle to the intruder regarding the ownship heading direction (θ), +the heading angle of intruder relative to ownship heading direction (ψ), the speed of the ownship +(vownship), and the speed of the intruder (vintruder). We can see a representation of the inputs in +Figure 14. +There are two extra parameters that are not used as inputs to the neural network. The first one +is the time until loss of vertical separation (τ), whereas the second one is the previous prediction +advice (aprevious). +These τ and aprevious parameters are discretized such that for each possible +combination of values for τ and aprevious, a different model is trained. This process resulted in a +total of 45 trained models, as mentioned before. Each trained model associates an input pattern +to five possible categories: clear of conflict (y0), weak left (y1), weak right (y1), strong left (y3), +and strong right (y4). Figure 15 contains a simple representation of a single the neural network +classifier associated with a value for τ and aprevious. Each neural network comprises 6 hidden layers, +28 + +Figure 14: Representation of the ACAS XU inputs. The black dot represents the position of the +ownship and the red dot the position of the intruder. ρ represents the Euclidean distance between +both aircrafts, θ the angle between the ownship heading direction and the vector that connects both +aircraft, and ψ the angle between the ownship and the intruder heading direction. +Figure 15: Representation of the inputs and the outputs of a single neural network from ACAS XU. +On the left of the neural network we present each of the five expected inputs: ρ, θ, ψ, vown and +vint. The neural network is defined according to a given τ and a aprev. To the right of the neural +network we have the expected probability for each output class. CC stands for clear of conflict, WL +for weak left, WR for weak right, SL for strong left, and SR for strong right. +containing 50 neurons each. +For further details on the training and prediction process of these +models, please refer to [1]. +Based on the trained models of the ACAS XU, ten desired properties have been proposed so that +this system works correctly according to its crafted design. In this paper, to compare with existing +verification approaches, we verified Property 1, formally stated as: +Property 1. The conditions are established as follows: +• input constraint: ρ ≥ 55947, 691, vownship ≥ 1145 and vintruder ≤ 60; +• output constraint: y0 ≤ 1500; +where y0 is the output associated with the clear of conflict output class. +29 + +intruder +ownship +Intruder +p +0 +OwnshipCC +p +WL +Neural network +WR +(t,a.. +prev +SL +own +SR +V +intRecall from the problem statement that the reachability analysis aims to verify a reachable set +R, obtained from X, such that R ∩ ¬Y = ∅ holds. Y denotes the expected output for the inputs in +X. For the Property 1, we have that: +X = {(ρ, θ, φ, vownship, vintruder) | +55947.691 ≤ ρ ≤ 60760, +−π ≤ θ ≤ π, +−π ≤ φ ≤ π, +1145 ≤ vownship ≤ 1200, +0 ≤ vintruder ≤ 60} +(33) +and that: +Y = {(y0, y1, y2, y3, y4) | y0 ≤ 1500} +(34) +Property 1 is the only one presented in this paper because only this property is evaluated by means +of comparison with other formal verification algorithms. However, all of the remaining properties +are formally described in [10]. +5.2 +Experimental description +We present in this section the procedures of our experimental setup. The experimental results +are divided into two main parts: the first part aiming to validate and compare the results with +algorithms from the literature, and the second part to evaluate the features of our approaches. For +the first part, we conducted the experiments by: +1. Generate the vertices of the input polytope, based on the input constraint imposed by Property +1; +2. Set a timeout of 24 hours for each algorithm verification; +3. Compute the output reachable set and the verification status for each reachability algorithm +(Exact polytope network mapping (EPNM), MaxSens [12], Ai2 [13] and ExactReach [11]); +4. Compute counterexamples and the verification status for the search algorithm (Reluplex [10], +Duality [16], MIPVerify [15] and NSVerify [14]); and +5. Compare results. +For the second experiment, which aims to analyze the parallelism behavior of the algorithm, the +experimental setup consists of: +1. Generate the vertices of the input polytope, based on the input constraint imposed by Property +1; +2. Set the number of parallel processes, denoted by p, such that p ∈ {1, 4, 8, 12, 16, 20, 24, 28, 32}; +and +3. Verify Property 1 for each p. +30 + +5.3 +Hardware and software specification +For comparative matters, we provide the specification for both hardware resources and software +language. The comparative experiment previously presented was performed in an Intel Xeon CPU +E5-2630 V4 of 2.20GHz, with 40 available CPU. Those algorithms from the literature and the +algorithms proposed in this work were developed in Julia language. The implementation of the +algorithms from the literature were available on [23]. +The implementation of our algorithms is +available on [24]. +5.4 +Comparative results +The validation and comparative results of the verification of ACAS Xu models for Property 1 +are presented in two different perspectives: firstly among those verification procedures that follow +reachability approaches, then comparing with approaches that make use of different techniques +(search or optimization). Finally, we present some useful features of the proposed approach. +5.4.1 +EPNM versus reachability approaches +By comparing the EPNM approach with those verification algorithms that follow a reachability +approach, as presented in Table 1, it can be seen that the proposed exact approach verified most of +the neural networks within the stipulated timeout time (43 out of 45 neural networks). As we can +see, none of the other existing exact approaches were able to verify a single model within a day of +execution (24 hours). Notice also that the approximate approaches (MaxSens and Ai2) finished their +execution, though, due to their over-approximation, these procedures did not estimate the correct +status well (which is acceptable, as these approaches are sound but not complete). +Table 1: Comparative results between the proposed approach (EPNM) regarding existing reacha- +bility approaches from the literature. These results refer to the verification of Property 1 of ACAS +XU models. +Proposed +Approaches from the Literature +Status +EPNM +ExactReach +MaxSens +Ai2 +holds +43 +- +- +- +violated +- +- +45 +45 +timeout +2 +45 +- +- +5.4.2 +EPNM versus search and optimization approaches +In comparison to optimization and search approaches, the proposed EPNM approach also reached +interesting results. +Compared to Reluplex results from the literature, EPNM could verify more +neural networks within the specified timeout. However, by executing Reluplex in the same hardware +conditions of EPNM, the verification was not completed within 24 hours. The same occurred for +the NSVerify procedure. +1Results extracted from [10]. +2Results from the authors for a 24-hour execution. +31 + +Table 2: Comparative results between the proposed approach regarding existing search and opti- +mization approaches from the literature. These results refer to the verification of Property 1 of +ACAS XU models. +Proposed +Approaches from the Literature +Status +EPNM +Reluplex1 +Reluplex (24 hours)2 +Duality +NSVerify +holds +43 +41 +- +- +- +violated +- +- +- +- +- +timeout +2 +4 +45 +- +45 +unknown +- +- +- +45 +- +5.4.3 +Parallel Computation +Due to the characteristics of the proposed approaches, their implementation allows the paral- +lelization in a procedural level. We chose to implement the parallelization in two of the procedures +(EI and II), because the remaining algorithms did not respond well due to the tradeoff between the +overhead and the speed-up of the parallelization. +The first one is the edge identification (EI) algorithm. For this procedure, we created a pool +for the execution with the size equal to the number of available threads. The pool guarantees that, +after each thread ends its execution, a new thread is started and takes the empty space. For each +vertex, a new thread was initiated, which verified if the adjacency property holds for the current +and each of the remaining vertices. As sharing memory was not necessary, because each thread has +its own adjacency list, no synchronization approach was implemented. By the end of the execution, +the algorithm concatenated the adjacency list calculated for each vertex. +The intersection identification (II) was the second parallelized procedure. Following the same +idea, a new thread was created for each vertex. After the initialization is completed, the procedure +identifies those intersection points between the current and each of its adjacent vertices. In this +case, similarly to the previous one, no synchronization was necessary, as each thread carried its own +list of intersections. At the end of the pool execution, the procured concatenated those intersection +points associated with each vertex. +The results of the second part of the experiments are depicted in Figure 16a. As can be seen in +this figure, as the number of available threads for the execution increases, the running time decreases +significantly. This characteristic of the algorithm can be explored for huge problems. +Figure 16b presents the speedup behavior for the algorithm EPNM. As expected, the algorithm +indeed reduce the runtime as there is an increment on the available threads, though the difference +between the real and the ideal curve indicates that the parallel processes are not ideally balanced +(there are threads waiting for some execution to end). +5.4.4 +Complexity behavior of the proposed approaches +Our experiments showed that, differently from the expected, the algorithm EPNM has a shorter +running time in comparison to APNM, for the ACAS XU model. This behavior can be explained +considering the total number of vertices that are processed in both algorithms. Figure 17 presents +the behavior of both, the total number of vertices and sets processes at each layer of a single neural +network from ACAS XU. +Figure 17a shows that, from the same input set, the algorithm APNM generates the greater +set of vertices for representing its approximation of the output set compared to both, EPNM and +PAPNM (i.e., 313286 vertices at the third layer). Figure 17b depicts the increment on the number +32 + +(a) The graph illustrates the execution +speed-up obtained from parallelizing the +proposed algorithm. +As the number of +available threads increases, there is a sig- +nificant reduction in the running time (du- +ration). +(b) Comparison between the actual speed +up of the EPNM and the ideal speed up. +Figure 16: Illustrative visualization of the EPNM parallelization behavior. We present both, the +runtime and the speed up for the algorithm EPNM execution on a single ACAS Xu model for the +property 1. +Table 3: Results on the running time of EPNM, APNM and PAPNM with d = 2 and d = 4 for the +first three layers of a single neural network. +(a) Behavior of the total number of vertices pro- +cessed for each layer of a neural network by APNM, +EPNM and PAPNM. +Layer +APNM +PAPNM +PAPNM +EPNM +(d = 2) +(d = 4) +1 +32 +32 +32 +32 +2 +323 +580 +580 +580 +3 +313286 +85404 +21977 +2502 +(b) Behavior of the total number of sets processed +for each layer of a neural network by APNM, EPNM +and PAPNM. +Layer +APNM +PAPNM +PAPNM +EPNM +(d = 2) +(d = 4) +1 +1 +1 +1 +1 +2 +1 +11 +21 +42 +3 +1 +78 +117 +149 +of sets for EPNM and PAPNM. Table 3 reports the data used to create Figure 17. +Both results lead to a lower average number of vertices across each of the sets for EPNM. On the +other hand, the opposite occurs to APNM which has a single set with a strongly increasing number +of vertices. Table 4 shows the average number of vertices per set for each algorithm at each layer +of the neural network. This means that the merging process (performed partially by PAPNM and +completely by APNM) induces a simplification on the total number of sets, though as a side effect +it significantly increases the total number of vertices to be processed. +The number of vertices within a set is directly related to the running time for each algorithm +because the computational complexity of each of the procedures that comprise APNM, PAPNM and +EPNM is directly related to the total of vertices processed. +33 + +Runtimexthreads +2.00×104 +1.50×104 +Runtime (s) +1.00×104 +5.00×103 +10 +15 +20 +25 +30 +threadsSpeedupxthreads +32 +Real +Ideal +28 +24 +20 +speedup +16 +12 +8 +4 +4 +8 +12 +16 +20 +24 +28 +32 +threads(a) Total of number vertices processed at +each layer of a single neural network. The +y-axis is in logarithmic scale. The APNM +algorithm presents a significantly higher in- +crement in the number of vertices after +layer 3, in comparison to PAPNM and +EPNM. Both PAPNM and EPNM execu- +tions have the same value at layer 2, as ex- +pected, though EPNM has a lower number +of vertices at layer 3. +(b) Total number of sets generated at each +layer processed by each of the algorithms. +As expected, APNM keeps 1 set at the end +of every layer execution. The EPNM has +the greatest increment on the number of +sets, which is an expected behavior. +Figure 17: Illustration of the complexity behavior of APNM, EPNM and PAPNM with d = 2 and +d = 4 for the first 3 layers of a single neural network from ACAS XU model. +Table 4: Average of the total number of vertices within each set for each algorithm. The results are +presented layer by layer. +Layer +APNM +PAPNM +PAPNM +EPNM +(d = 2) +(d = 4) +1 +32 +32 +32 +32 +2 +323 +52.7 +27.6 +13.8 +3 +313286 +1094.9 +187.8 +16.8 +6 +Conclusion +In this work, we proposed two vertex-based reachability algorithms for formal verification of +deep neural networks. These algorithms compute a reachable output set, which may consist of a +set of polyhedral sets, for a given input polyhedral set, satisfying different properties: the first one +(APNM) computes an approximation for the output reachable set, while the second one (EPNM) +computes the exact output reachable set. +Supported by formal demonstrations of correctness, the proposed algorithms were shown to +correctly verify properties of neural networks with the ReLU activation function. More specifically, +it was shown that APNM yields an overestimation of the output reachable set, while EPNM computes +34 + +VerticesxLayer +EPNM +PAPNM (d=2) +PAPNM (d=4) +105.0 +APNM +104.5 +104.0 +Vertices +103.5 +103.0 +102.5 +102.0 +10l.5 +1 +2 +m +LayerSets x Layer +150 +EPNM +PAPNM (d=2) +135 +PAPNM (d=4) +APNM +120 +105 +90 +Sets +75 +60 +45 +30 +15 +0 +2 +m +Layerthe exact reachable set. +Our proposal was applied to a benchmark problem for neural network verification and compared +to some of the algorithms previously proposed in the literature. The results showed that among +the verification algorithms that make use of reachability analysis, the presented EPNM approach +concluded most of the verifications (43 out of 45 neural networks), differently from the ExactReach, +which is another exact approach from the literature that could not verify any neural network within +the specified timeout. Compared to those algorithms that are not complete, despite the fact that +these approaches were able to finish their executions, the expected output was not reached in any +case (see Table 1). +In comparison to those methods that make use of optimization and search strategies, the result +reported in the literature for Reluplex surpassed ours in terms of running time. However, under the +same hardware conditions, our approach was able to overcome their results. +Finally, we argue that the algorithms proposed in this paper are strong candidates for the ver- +ification of neural networks with ReLU activation. Additionally, the running time can be reduced +drastically by using multiple processing cores. Future work includes the investigation of a different +construction for the search approach in EPNM and of a third approach that can be designed to +improve performance by using heuristics for the reduction of sets and vertices during the search +process. +Acknowledgment +This work was funded in part by Funda¸c˜ao de Amparo `a Pesquisa e Inova¸c˜ao do Estado de Santa +Catarina (FAPESC) under grant 2021TR2265. +References +[1] K. D. Julian, M. J. Kochenderfer, and M. P. Owen, “Deep neural network compression for +aircraft collision avoidance systems,” Journal of Guidance, Control, and Dynamics, vol. 42, +no. 3, pp. 598–608, 2019. +[2] C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. J. Goodfellow, and R. Fergus, +“Intriguing properties of neural networks,” CoRR, vol. abs/1312.6199, 2014. +[3] I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing adversarial examples,” +CoRR, vol. abs/1412.6572, 2015. +[4] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu, “Towards deep learning mod- +els resistant to adversarial attacks,” in International Conference on Learning Representations, +ICLR 2018, 2018. +[5] F. Tram`er, A. Kurakin, N. Papernot, I. Goodfellow, D. Boneh, and P. McDaniel, “Ensemble +adversarial training: Attacks and defenses,” in International Conference on Learning Represen- +tations, ICLR 2017, 2017. +[6] Y. Song, T. Kim, S. Nowozin, S. Ermon, and N. Kushman, “Pixeldefend: Leveraging generative +models to understand and defend against adversarial examples,” CoRR, vol. abs/1710.10766, +2017. +35 + +[7] K. Grosse, P. Manoharan, N. Papernot, M. Backes, and P. McDaniel, “On the (statistical) +detection of adversarial examples,” CoRR, vol. abs/1702.06280, 2017. +[8] J. H. Metzen, T. Genewein, V. Fischer, and B. Bischoff, “On detecting adversarial perturba- +tions,” in Proceedings of 5th International Conference on Learning Representations (ICLR), +2017. +[9] J. Lu, T. Issaranon, and D. Forsyth, “Safetynet: Detecting and rejecting adversarial examples +robustly,” in Proceedings of the IEEE International Conference on Computer Vision, pp. 446– +454, 2017. +[10] G. Katz, C. Barrett, D. L. Dill, K. Julian, and M. J. Kochenderfer, “Reluplex: An efficient +SMT solver for verifying deep neural networks,” in International Conference on Computer +Aided Verification, pp. 97–117, Springer, 2017. +[11] W. Xiang, H.-D. Tran, and T. T. Johnson, “Reachable set computation and safety verification +for neural networks with ReLU activations,” arXiv preprint arXiv:1712.08163, 2017. +[12] W. Xiang, H.-D. Tran, and T. T. Johnson, “Output reachable set estimation and verification +for multilayer neural networks,” IEEE Transactions on Neural Networks and Learning Systems, +vol. 29, no. 11, pp. 5777–5783, 2018. +[13] T. Gehr, M. Mirman, D. Drachsler-Cohen, P. Tsankov, S. Chaudhuri, and M. Vechev, “Ai2: +Safety and robustness certification of neural networks with abstract interpretation,” in IEEE +Symposium on Security and Privacy (SP), pp. 3–18, IEEE, 2018. +[14] A. Lomuscio and L. Maganti, “An approach to reachability analysis for feed-forward relu neural +networks,” arXiv preprint arXiv:1706.07351, 2017. +[15] V. Tjeng, K. Xiao, and R. Tedrake, “Evaluating robustness of neural networks with mixed +integer programming,” arXiv preprint arXiv:1711.07356, 2017. +[16] K. Dvijotham, R. Stanforth, S. Gowal, T. A. Mann, and P. Kohli, “A dual approach to scalable +verification of deep networks,” in the Conference on Uncertainty in Artificial Intelligence (UAI), +vol. 1, p. 3, 2018. +[17] E. Wong and Z. Kolter, “Provable defenses against adversarial examples via the convex outer ad- +versarial polytope,” in International Conference on Machine Learning, pp. 5286–5295, PMLR, +2018. +[18] X. Huang, M. Kwiatkowska, S. Wang, and M. Wu, “Safety verification of deep neural networks,” +in International Conference on Computer Aided Verification, pp. 3–29, Springer, 2017. +[19] P. McMullen and E. Schulte, Abstract regular polytopes, vol. 92. Cambridge University Press, +2002. +[20] I. Z. Emiris, V. Fisikopoulos, and B. G¨artner, “Efficient edge-skeleton computation for polytopes +defined by oracles,” Journal of Symbolic Computation, vol. 73, pp. 139–152, 2016. +[21] A. Schrijver et al., Combinatorial Optimization: Polyhedra and Efficiency, vol. 24. Springer, +2003. +[22] A. Schrijver, Theory of Linear and Integer Programming. John Wiley & Sons, 1998. +36 + +[23] C. Liu, T. Arnon, C. Lazarus, C. Strong, C. Barrett, M. J. Kochenderfer, et al., “NeuralVerifi- +cation.jl,” 2019. +[24] J. Zago, E. Camponogara, and E. Antonelo, “vertexBasedRechabilityAnalysis.jl,” 2023. +37 + diff --git a/INFLT4oBgHgl3EQfIy9R/content/tmp_files/load_file.txt b/INFLT4oBgHgl3EQfIy9R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..11722fd6b66bbdc5c4eb3778d2f7be4684048ba1 --- /dev/null +++ b/INFLT4oBgHgl3EQfIy9R/content/tmp_files/load_file.txt @@ -0,0 +1,1064 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf,len=1063 +page_content='Vertex-based reachability analysis for verifying ReLU deep neural networks Jo˜ao Zago∗, Eduardo Camponogara†, Eric Antonelo‡ January, 2023 Abstract Neural networks achieved high performance over different tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' image identification, voice recognition and other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Despite their success, these models are still vulnerable regarding small perturbations, which can be used to craft the so-called adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Different approaches have been proposed to circumvent their vulnerability, including formal verification systems, which employ a variety of techniques, including reachability, optimization and search procedures, to verify that the model satisfies some property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In this paper we propose three novel reachability algorithms for verifying deep neural networks with ReLU activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The first and third algorithms compute an over-approximation for the reachable set, whereas the second one computes the exact reachable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Differently from previously proposed approaches, our algorithms take as input a V-polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Our experiments on the ACAS Xu problem show that the Exact Polytope Network Mapping (EPNM) reachability algorithm proposed in this work surpass the state-of-the-art results from the literature, specially in relation to other reachability methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 1 Introduction Regardless of the success of deep neural networks in computer vision and natural language processing, these models are susceptible to small perturbations applied to their inputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' it is possible to misguide the model output by applying a designed perturbation to a given input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For instance, the ACAS Xu model [1] (explained later in detail), that responded differently from expected while under specific circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The inputs purposefully designed to force a misbehavior of the neural network are denoted as adversarial examples [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' To overcome such vulnerability many different approaches have been previously employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' One proposed the application of algorithms that were able to generate adversarial examples, the so called adversarial attacks, and subsequently applied these inputs in the training process of the network [2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' There were also those approaches that aim to identify the adversarial examples before feeding them as input to the neural network [6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Even though these procedures helped to reduce the vulnerability of the neural networks, these models remained vulnerable to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' ∗joao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='zago@posgrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='ufsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='br †eduardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='camponogara@ufsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='br ‡eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='antonelo@ufsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='br 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='12001v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='LG] 27 Jan 2023 Formal methods were also applied to certify or guarantee that the model behaves as expected under some circumstances or within a specified domain region, nevertheless [10] showed that the verification problem is NP-hard, leaving the process of certifying large models still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The existing formal procedures can be classified into three different categories: 1) reachability methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2) optimization methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' and 3) search methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The first one relies on calculating the output mapping of an input set [11, 12, 13], the second one comprises the application of math- ematical optimization (Mixed Integer-Linear Programming or Convex Optimization) to identify counter-examples [14, 15, 16, 17], and the third makes use of both reachability and optimization approaches in conjunction with search methods for identifying counter-examples [10, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In this paper, we propose three novel reachability algorithms: APNM and PAPNM algorithms that compute an over-approximation for the output, while EPNM which computes the exact map- ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We present demonstrations on the behavior and correctness of these algorithms and case studies of their applications for comparison with existing algorithms from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We also show that the algorithms proposed in this work are highly parallelizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The rest of this paper is organized into five sections: Section 2 gives an overview of the existing algorithms and related works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Section 3 describes the proposed algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Section 4 addresses the demonstrations regarding the completeness and soundness of the proposed algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Section 5 presents a study case for application and comparison;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' and finally Section 6 discusses the outcome of the procedures presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2 Related Works Formal verification of neural networks is receiving a huge amount of attention mainly because of its importance in security sensitive tasks such as the application of neural networks in autonomous vehicles, systems controllers, aeronautics, and several other applications that can possibly involve financial, human or environmental injury [18, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As previously presented, there are three main research areas developed to provide guarantees for neural networks: 1) reachability methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2) optimization methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' and 3) search methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In this work, we propose three novel approaches regarding reachability analysis, which consist of two major steps: computing the output set by mapping a subset of the neural network domain and comparing the output set with the specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' One of the algorithms developed in the literature to verify neural networks by means of reach- ability analysis is denoted as MaxSens [12], which computes an over-approximation to the output reachable set and compares it to the desired specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As this algorithm computes an over- approximation it does not satisfy the completeness property of a verification algorithm, although it is sound (if the property is verified then the algorithm guarantees that it is satisfied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' To compute such an over-approximation, MaxSens divides the input set into several smaller hyperrectangles and computes the maximum sensitivity for each of them layer-by-layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The output of this algorithm consists of the union of several hyperrectangles, which approximates the exact reachable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Note that their approach can be applied to neural networks with any activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Another approach from the literature, denoted as ExactReach [11], computes the exact reachable set given an input set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This procedure takes as inputs a convex H-polytope (a polytope represented by its inequalities) and computes the exact mapping layer-by-layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The authors proposed this approach specifically for the ReLU activation function, which is reasonable as this function has achieved promising results for convolutional and fully connected neural networks, which are widely applied in different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Due to the nature of the ReLU activation function, the authors separated the non-linear mapping process in three cases: 1) all the elements of the input are positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2 2) all the elements of the input are negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' and 3) the input has positive, negative or null elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' These cases cover all the mapping possibilities regarding the ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Similarly to the aforementioned approaches, we propose in this paper three novel algorithms for reachability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' However, instead of using the H-polytope representation, each of our proposed procedures take as input set a V-polytope (a polytope represented by its vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We demonstrate the correctness of both algorithms and compare them with the literature (not only with those that make use of reachability analysis) by using a case study (ACAS Xu [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3 Algorithms In this section we will state the verification problem and the three algorithms proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The demonstrations regarding the correctness of each procedure will be presented in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='1 Problem statement Let F : Rn �→ Rm represent a mapping given by a neural network composed of L layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Then, for a given x ∈ Rn, F(x) = (FL ◦ FL−1 ◦ · · · ◦ F2 ◦ F1)(x), where Fl is the mapping given by the l-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Further, Fl consists of the composition of an affine mapping and a non-linear mapping (in this case a ReLU function): Fl(xl) = ReLU(Wlxl + θl), where Wl and θl denote the weight matrix and bias vector of the l-th layer, respectively, and xl is the input to the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Suppose that X ⊆ Rn is the input set that we want to verify and R = {F(x) | x ∈ X} is the exact output set associated with X, which resulted from the application of the neural network F to each input in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Moreover, the set Y comprises the expected output set for the inputs from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Then, the verification problem consists of assuring that: R ∩ ¬Y = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' (1) In other words, we want to guarantee that there will not exist an input of F in X that will cause the network to generate an undesired output (an output that is not in Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' To perform such a verification, one needs to start by calculating the output set associated with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As we presented in previous sections, there are approaches that compute the exact reachable set and other approaches that over-approximate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For those that compute an approximation for the output set, denoted by �R, we will have that R ⊆ �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Then, we can see that if �R ∩ ¬Y = ∅, then the property given by the Equation 1 will still be assured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In the following sections we will present the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='2 Approximate Polytope Network Mapping (APNM) The first algorithm proposed in this work is called Approximate Polytope Network Mapping (APNM) and consists of a procedure to compute an over-approximation for the reachable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Let P be a convex closed polytope, defined as a convex combination of its vertices, namely P = � o � i=1 λivi ���� o � i=1 λi = 1, λi ≥ 0 and vi ∈ V, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} � , (2) where V is the set of vertices of P (V = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , vo}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' To achieve its goal, the algorithm has five parts: 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Affine map of the vertices with weights and biases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Adjacent vertex identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Polytope intersection with an orthant’s hyperplanes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' ReLU mapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Removing non-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In what follows, the way the algorithm works will be explained with a simple 2-dimensional problem for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The input set P is presented in Figure 1: the set of vertices is V = {v1, v2, v3, v4}, where v1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0), v2 = (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0), v3 = (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0) and v4 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' and the set of edges is E = {(v1, v2), (v1, v4), (v2, v3), (v3, v4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 1: The input set for visualizing each step of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Affine map of the vertices with weights and biases The affine map comprises the product of each vertex of P by the weights associated with layer l plus the biases of the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Algorithm 1 contains the steps to compute the affine map of all vertices from V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Algorithm 1: Affine Map (AM) Input: V ∈ Ro×n, V = [v[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , v[o]], v[i] ∈ Rn , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o}, W ∈ Rm×n and θ ∈ Rm Function AM(V, W, θ): let Z[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='.o] be a new array, where each Z[i] is an empty array, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} do Z[i] = Wv[i] + θ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' end return Z 4 2 V2 V1 1 0 1 2 2 1 0 1 2 1By applying the affine map to the input set, presented in Figure 1, we have the output of this step as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The weight matrix W and biases θ employed in this toy example are as follows: W = �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='492693 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='29232 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='925861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='675146 � , θ = �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='18857972 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='14839205 � (3) Note that, as expected, the output of the current operation is a simple affine transformation of the vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 2: Representation of the output of the affine map operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Adjacent vertex identification Following the affine map, the edge identification process takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Let vi, vj ∈ V be two distinct vertices, such that i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' If there is at least one combination of the vertices of V, except from vi or vj, that allow us to compute the middle point of vi and vj, given by (vi + vj)/2, then vi and vj are not adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Equation 4 presents this feasibility problem as a MILP (Mixed Integer- Linear Programming), where λ ∈ Ro is the vector used to represent the convex combination of the 5 2 V1 1 0 1 V3 2 2 1 0 1 2 1vertices of P and η ∈ {0, 1} is the binary variable that enables evaluating vi or vj separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' max 0 , (4a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' (vi + vj) 2 = o � k=1 vkλk , (4b) o � k=1 λk = 1 , (4c) λk ≥ 0 ,∀k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} , (4d) λi ≤ η , (4e) λj ≤ 1 − η , (4f) η ∈ {0, 1} (4g) Algorithm 2 presents the identification of adjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' It creates an undirected graph using an adjacency list as the data structure to represent the 1-skeleton of the polytope, which is the edge structure of some polytope [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We denote the adjacency list of the undirected graph by E, were each element is a set containing the indices of the adjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Algorithm 2: Edge-skeleton Identification (EI) Input: V ∈ Ro×n, V = [v[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , v[o]], v[i] ∈ Rn , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} Function EI(V): let E[1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' o] be a new array, where each E[i] , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o}, corresponds to an empty set for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} do for j ∈ {i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} do if ¬∃λ ∈ Ro : 1 2(v[i] + v[j]) = �o k=1 v[k]λ[k] ∧ �o k=1 λ[k] = 1 ∧ λ[k] ≥ 0 , ∀k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} ∧ (λ[i] = 0 ∨ λ[j] = 0) then E[i] ← E[i] ∪ {j} end end return E Following the toy problem, where the affine map is shown in Figure 2, the undirected graph that represents the edge structure of the affine map of P is presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This graph G = (V, E) consists of the vertex set V = {v1, v2, v3, v4} and edge set E = {(v1, v2), (v1, v4), (v2, v3), (v3, v4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Notice that each edge in the undirected graph indicates that the associated vertices are connected by an edge of the corresponding polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Polytope intersection with orthant’s hyperplanes The next step for the algorithm concerns the identification of those points at the intersection of the hyperplanes that define each orthant with the edges of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As the edges of the polytope were previously computed, only the edges whose extreme points belong to different orthants need 6 Figure 3: Undirected graph that represents the edge structure of the output of the affine map of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' to be identified, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=', extreme points that have at least one component with a different sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' So, we compute the difference of the sign for each element of both extreme points for a given edge, given by Equation 5: a = sign(vi) − sign(vj) (5) where i denotes the vertex under consideration, j ∈ Ei represents the index of the vertices that are adjacent to vi, and sign : Rn → Rn is the signal mapping that associates 1 for the positive elements, −1 for the negative elements, and 0 for the null ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We also define a function σ : R → R, given by Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the cases where a component of a is greater than or equal to 2, we have that the sign indeed changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For those cases where the difference is equal to 1, there was a null component in one of the vertices, which does not indicate that they are in different orthants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' σ(α) = � 1, if |α| ≥ 2 0, otherwise (6) Then, for those cases where there is a sign change, or, in other words, for those cases that satisfy the condition: n � k=1 σ(ak) ̸= 0 (7) we compute those points at which the intersection between the edge and the hyperplane occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Each sign change generates an intersection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Given two vertices vi and vj, suppose that there is a k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n} for which σ(ak) ̸= 0, indicating that vi and vj belong to different orthants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' So, there exists a value of λ, such that 7 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='00 ≤ λ ≤ 1, defining a convex combination of the vertices that lies on the orthant hyperplane, namely a λ that satisfies: λvj,k + (1 − λ)vi,k = 0 ⇐⇒ λ = −vi,k vj,k − vi,k (8) Figure 4 contains a visual representation of this process for the case of vertices v1 = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='988207, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='45261) and v4 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='59643, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='102323).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Notice that a = sign(v1) − sign(v4) = (−1, 1) − (1, 1) = (−2, 0), thus σ(a) = (1, 0) implying that v1 and v4 lie in different orthants, and the edge connecting them intercepts the hyperplane defining the respective orthant at a point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' By using the above equation, we compute the convex combination parameter λ to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='382338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 4: Representation of the intersection identification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The blue dots represent two adjacent vertices and the line segment is the edge connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The red dot is the intersection point of the edge with respect to the orthant hyperplane supporting plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Then, we compute each component s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n} of the point p ∈ Rn, as presented by Equation 9, ps = (vj,s − vi,s)λ + vi,s, (9) which represents the intersection between an edge of P and one of the hyperplanes of an orthant from Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the example in Figure 4, the intercept of the edge (v1, v4) with the supporting hyperplane {x = (x1, x2) : x1 = 0} of the orthant {x : x ≥ 0} is p = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='93635).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The procedure, formalized by Algorithm 3, computes the intersection points between each edge of P and each orthant’s supporting hyperplane that are intercepted by the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The result of the intersection identification for the output of the affine map of P is presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' There we present all the vertices of the polytope in addition to the computed intersection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' ReLU Mapping After calculating the intersection points between edges and orthant’s hyperplanes, the ReLU mapping of the resulting points is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We apply the mapping ReLU : Rn → Rn, defined by Equation 10, to all the points given by the previous procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Algorithm 4 summarizes the process 8 2 1 V 0 1 2 2 1 0 1 2 1Algorithm 3: Orthant intersection identification (II) Input: V ∈ Ro×n where V = [v[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , v[o]], v[i] ∈ Rn , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o}, E = [E[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , E[o]] Function II(V, E): for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} do /* For each edge of vertice i / for j ∈ E[i] do a ← sign(v[i]) − sign(v[j]) for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n} do if σ(a[k]) ̸= 0 then λ ← −v[i,k] v[j,k]−v[i,k] let p be a point in Rn for s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n} do p[s] ← v[j,s]−v[i,s] v[i,s] λ end V ← [V | p] /* Append p to matrix V / end end end return V Figure 5: Intersection identification results representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' where V is the output vertex set from Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' ReLU(x) = max(0, x) (10) We can see the result of the application of the ReLU mapping as presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As expected, all the vertices were projected to the positive orthant, whereby vertices v2, v5, and v6 are projected onto the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Notice that the output set is not convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 9 2 V1 1 VA V: 0 NB 1 V6 V3 2 2 1 0 1 2 1Algorithm 4: ReLU map Input: V ∈ Ro×n, V = [v[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , v[o]], v[i] ∈ Rn , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} Function ReLU(V): for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} do for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n} do v[i, j] ← max(0, v[i, j]) end end return V Figure 6: Representation of the output of the ReLU mapping of the vertices and intersection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Removing Non-vertices To simplify the output of a single layer, algorithm APNM computes the convex hull on the output of the ReLU mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' To compute the convex hull, the algorithm removes the points that are not vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' To create a generalized process for eliminating non-vertices, and due to the high dimensionality of the internal layers of the neural network, we propose the application of a feasibility analysis to identify the points that can be expressed as a convex combination from the remaining points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 10 2 V 1 V 0 V5 V3 1 2 2 1 0 1 2 1The feasibility problem is formalized in Equation 11, as follows: min 0 , (11a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' o � i=1 λivi = vk , (11b) o � i=1 λi = 1 , (11c) λk = 0 , (11d) λi ≥ 0 , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o}, (11e) where, for each vertex candidate, denoted by vk, we search for a λ vector such that vk can be described as a convex combination of the remaining vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' If that is the case, then vk is not a vertex and it can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Otherwise, the point is a vertex and must remain in the set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The non-vertex elimination is formalized in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Algorithm 5: Removing Internal Points (RP) Input: V ∈ Ro×n, V = [v[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , v[o]], v[i] ∈ Rn , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} Function RP(V): for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} do if ¬∃λ ∈ Ro : �o i=1 λiv[i] = v[k] ∧ �o i=1 λi = 1 ∧ λk = 0 ∧ λi ≥ 0 , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} then V ← [v[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , v[k − 1], v[k + 1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , v[o]] end return V The output of the process of removing non-vertices can be viewed in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As we can see, instead of returning the exact non-convex set, this algorithm computes an over-approximation for the output set which is simpler than the exact output as it is a unique set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Approximate Polytope Network Mapping Finally, we show the complete process, presented by the Algorithm 6, to compute an approxima- tion for the output reachable set for a given input polyhedron P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='3 Exact polytope network mapping (EPNM) We present in this section the second proposed approach, which similarly to the first one computes the reachable set by using the vertices of a given input polytope, but computes the exact output instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We consider the same input closed convex polytope P and the same set of vertices V as presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This approach comprises six parts: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Affine map of the vertices with weights and biases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Adjacent vertex identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Origin verification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 11 Figure 7: Representation of the convex hull of the output of the ReLU mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Algorithm 6: Simple Approximated Politope Network Mapping Input: V ∈ Ro×n, Wl and θl for l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , L} Function SAPNM(V, W, θ): Z[0] ← V for l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , L} do ˆZ[l] ← AM(Z[l − 1], Wl, θl) /* Compute affine mapping for layer l / if l < L then E[l] ← EI(ˆZ[l]) /* Compute edge-skeleton of polyhedron given by vertices ˆZ[l] / ˆZ[l] ← II(ˆZ[l], E[l]) /* Obtain intercept of edges with horthant hyperplanes / ˆZ[l] ← ReLU(ˆZ[l]) /* Apply ReLU mapping to the vertex set ˆZ[l] / Z[l] ← RP(ˆZ[l]) /* Remove non-vertex points from vertex set Z[l] / end return Z[L] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Polytope intersection with orthant’s hyperplanes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Separate points according to the orthant to which they belong;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' ReLU mapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Removing non-vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In this section we only present parts 3 and 5, which were not previously introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Origin verification After we compute the edges of P, we need to verify if the origin is inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In case it is true, then we need to insert it into the set of vertices that will be partitioned with respect to the orthant that 12 2 V 1 VA 0 V2 1 2 2 1 0 1 2 1they belong in the next part of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This verification is compulsory due to the separation of vertices according to the orthant to which they belong (for instance, if the origin is in P and is not in the set of the vertices of the partitions of the polytope, then the union of the partitions will not be equal P), as presented in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Notice that, each partition represent the portion of P that is inside an orthant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=', the portion of P that is in the positive orthant, in Figure 8, is the union of the red area and the shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We can see that if the origin is not part of the set of vertices of each partition of P, after the orthant separation process, the shaded area will be missing for the portion of P inside the positive orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Thus, we need to include the origin point in the respective set so that the portion of P in the positive orthant also includes the shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' To Figure 8: Representation of the issue that will occur if the origin is not inserted to the vertex set for the orthant separation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' perform such a verification, we state the search problem as a feasibility analysis, in which we try to verify if there exists a vector λ ∈ Ro such that the origin is a convex combination of the vertices in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This problem is formalized by Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' min 0 , (12a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' o � i=1 λivi = 0 , (12b) o � i=1 λi = 1 , (12c) λi ≥ 0 , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' (12d) Algorithm 7 formalizes the search process associated with the verification of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 13 V 2 V V 3 X V 6 V 5Algorithm 7: Origin Search (OS) Input: V ∈ Ro×n, V = [v[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , v[o]], v[i] ∈ Rn , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} Function OS(V): if ∃λ ∈ Ro : �o i=1 λivi = 0 ∧ �o i=1 λi = 1 ∧ λi ≥ 0 , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} then V ← [v[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , v[o] | 0] /* Adding the origin to the vertex set / return V Separating points to their respective orthants We present in this section the process of splitting the vertex set V into several sets, each associated with a different orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In other words, we split V in such a way that each resulting set contains points located in a single orthant (observe that the convex hull of each set of vertices represents the partition of P that is inside an orthant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We begin by computing: b = 1 2(sign(vi) + 1|vi|) (13) where, vi is the vertex under analysis, sign : Rn → Rn is the sign mapping previously presented and 1|vi| is a vector with all the components equal to one and has size |vi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The elements of b with value equal to 1 are associated with a positive element of vi, the elements equal to 0 are associated with negative components of the vertex, and those components of b equal to 1/2 are associated with the null components of the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Notice that a null component of a vertex means that this point belongs to at least two different orthants (the origin being a special case inside all the orthants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For those cases where the component of b is equal to 1/2, we must guarantee that the associated vertex vi is properly inserted into each of those sets that are associated with the orthants it belongs to (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=', if vi has one null component, it must be inserted in two sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Then, the algorithm starts a verification process to identify those component of b that are equal 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In case it is equal true, two copies of vi must be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This verification process is repeated until all components of b have been checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This process is detailed in Algorithm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' After checking for null values in vi, the sets in which it must be placed must be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' It should be noted that at the end of this process, b is a binary vector that represents the index in which vi must be placed (there might be more than one b for a single vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Equation (14) presents the process for converting b into a decimal number q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The calculation of the index is presented in Algorithm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' q = n � i=1 bi2i−1 (14) Figure 9 illustrates a toy example of how this process works for a single vertex vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In this example, vi = (−2, 0), with the second component being null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As a result, b = (0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Next, two copies of b are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The components of the first (second) copy that are equal to 1 2 are replaced by 0 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In this case, as there was only one null component, the generated vectors are (0, 0) and (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The vector generated from b contains the binary representation of the indices of the sets that vi must be placed in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In this example, vi must be placed in the orthants with indices 0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' It is important to note that the indices used by the algorithm to identify the destination sets are different from the traditional orthant enumeration (2nd and 3rd quadrants in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The orthant separation complete process is computed accordingly by Algorithm 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Continuing the example from the previous subsection, the orthant separation performed by Algorithm 10 will divide the polytope illustrated in Figure 5 into four separate polytopes, one 14 Algorithm 8: Zeros Verification (ZV) Input: b ∈ [0, 1]n Function ZV(b): A ← {b} B ← {b} while |B| ̸= 0 do B ← ∅ for a ∈ A do for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , |a|} do /* Check for those components that are equal 1/2 / if a[i] = 1/2 then b ← a b[i] ← 1 B ← B ∪ b /* Insert the first copy into B / b ← a b[i] ← 0 B ← B ∪ b /* Insert the second copy into B / break end end if |B| ̸= 0 then A ← B end return A /* return a set of vectors / Algorithm 9: Get Array Position (AP) Input: b ∈ {0, 1}m Function AP(b): let q be an integer q ← 0 for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n} do q ← q + b[i]2i−1 end return q for each orthant, as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' It is worth noting that vertices v5, v6, v7, v8, and v9 have been placed in more than one set, as they are located in more than one orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Exact Polytope Network Mapping The layer mapping and the complete process for the Exact Polytope Network Mapping is formal- ized in Algorithm 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The algorithm takes as input the vertices of the input polytope aimed to be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Then, it computes the affine map of these vertices (calculated by the algorithm AM) and identifies the edges between adjacent vertices (with the algorithm EI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the vertices connected by an edge, the algorithm computes the intersection of the corresponding edge with each orthant’s supporting hyperplane, when those vertices are not in the same orthant (by means of the algorithm 15 Figure 9: Vertex separation process for a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Given a vertex vi = (−2, 0), the associated b is given by (0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As we have that there is a component of b equal to 1 2, we split it into two vectors in such a way that the component with value equal to 1 2 is substituted by 1 and 0 in the newly created vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' These new vertices contain the binary representation of the orthant’s index that vi belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In the presented example, vi belong to 2 different orthants (the blue and the green ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Algorithm 10: Separate points per orthant (SP) Input: V ∈ Ro×n, V = [v[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , v[o]], v[i] ∈ Rn , ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} Function SP(V): let Z[1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2n] be an array for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n} do let b[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n] be an array b ← 1 2(sign(v[i]) + 1|v[i]|) % identify if there are null coordinates B ← ZV(b) /* Identify the orthants to which v[i] belongs / for b′ ∈ B do q ← AP(b′) /* Calculate index position of orthant b′ to which v[i] belongs / if Z[q] = ∅ then Z[q] ← v[i] else Z[q] ← [Z[q] | v[i]] /* Appending the new vertex / end end return Z II), and verifies whether or not the origin belongs to the polytope under analysis (checked by the algorithm OS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Next, it separates all of these points in different sets, where each set contains those vertices that belong to a single orthant (computed by the algorithm SP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Finally, the algorithm per- forms the ReLU mapping and removes non-vertices for each partition generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As later presented, 16 X2 v, = (-2,0) b = (0,/2) V, = (-2,0) (0,0) (0,1) X1 q = 0*21+0*2° = 0 q = 1*21+0*2° = 2 (3rd orthant) (2nd orthant)Figure 10: Representation of the separation of the polytope P after the application of the affine map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' the ReLU mapping preserves the convexity inside a given orthant, though there are points that are not vertices after the application of the non-linear mapping (check vertex v3 in Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Finally, after the application of the ReLU mapping and the removal of non-vertices (RP) for each partition in the set, as illustrated in Figure 10, the result of the EPNM algorithm for a single layer mapping is shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Observe that this output, which is a union of sets, represents the mapping of a unique layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For each of the sets that result from the previous layer, the algorithm computes the exact mapping for the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This process will repeat until the algorithm maps all the sets to the last layer of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For instance, considering that the visual problem has one extra layer, each of the four output sets (one set for each orthant, as the intersection of P with each orthant is not empty) will be mapped to the next layer following the same described process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='4 Partially approximated polytope network mapping (PAPNM) We have proposed a third approach, which involves a slight modification of the EPNM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Instead of an exact mapping between layers, this approach allows for the merging of some of the resulting sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' By computing the convex hull of the union for some of the output sets, we aim to achieve a more accurate approximation of the output set (compared to APNM) while also reducing the execution time of the algorithm (compared to EPNM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The result is presented in Algorithm 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The main difference between Algorithm 11 and Algorithm 12 is the application of the MS procedure after the separation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The MS algorithm comprises the merging procedure, where a set of sets of vertices is given as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As each set of vertices represents a single polytope, this procedure merges some of these sets of vertices to produce fewer sets compared to the exact mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' It is worth noting that the output of MS is an overapproximation of the input set it receives, and that the extreme case in which a single set of vertices is computed is exactly the case of APNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Here, we propose a basic approach for the merging process in Algorithm 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The MS algorithm takes the set P which contains the sets of vertices and an integer d as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The sets of vertices are then grouped into sets of size d and each group is merged (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='e, for the case where P has 13 sets 17 2 V1 1 VA V 0 NS 1 V6 V3 2 2 1 0 1 2 1Algorithm 11: Exact Polytope Network Mapping Input: V ∈ Ro×n, /* vertices of the polytope / Wl and θl for l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , L} /* network layers’ weights and biases / Function EPNM(V, W, θ): let P be an empty set P ← {V} /* The list of polytopes’s vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' starting with the input polytope / for l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' L} do ˆP ← ∅ /* ˆP is the auxiliary set that represents P in the next layer */ for Z ∈ P do /* iterate for each set Z of vertices in P / if l > 1 then Z ← ReLU(Z) /* Apply ReLU mapping to the vertex set Z / Z ← RP(Z) /* Remove non-vertex points from vertex set Z / Z ← AM(Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Wl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' θl) /* Perform affine mapping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Z ← WlZ + θl / if l < L then E ← EI(Z) /* Compute edge-skeleton E of the polyhedron given by vertices Z / Z ← II(Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' E) /* Add edge intercepts with orthant’s supporting hyperplanes to vertex set / Z ← OS(Z) /* Add origin to the vertex set if needed / Z ← SP(Z) /* Transform the vertex set into a list of vertices for each orthant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' considering the orthants to which each vertex belongs / for k ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , |Z|} do if Z[k] ̸= ∅ then ˆP ← ˆP ∪ Z[k] end end P ← ˆP end return P and d = 3, there will be 4 groups of 3 sets and 1 group of 1 set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' It is important to note that this is just a simple example of the merging procedure and that different heuristics can be implemented to improve this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The use of this merging procedure allows for different levels of approximation, resulting in a range of possible approximations from the exact mapping (EPNM) to the coarser case (APNM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' It is important to ensure that the merging procedure returns an overapproximation of the exact mapping of each layer, which is a necessary condition to ensure the soundness of PAPNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 4 Demonstrations We present in this section demonstrations that provide theoretical guarantees for the correctness of each proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 18 Algorithm 12: Partially approximate polytope network mapping Input: V ∈ Ro×n, /* vertices of the polytope / Wl and θl for l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , L} /* network layers’ weights and biases / /* d is the size of each group of sets that will be merged / Function PAPNM(V, W, θ, d): let P be an empty set P ← {V} for l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , L} do ˆP ← ∅ for Z ∈ P do if l > 1 then Z ← ReLU(Z) Z ← RP(Z) Z ← AM(Z, Wl, θl) if l < L then E ← EI(Z) Z ← II(Z, E) Z ← OS(Z) Z ← SP(Z) Z ← MS(Z, d) /* new function for merging sets / for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , |Z|} do ˆP ← ˆP ∪ Zk end end P ← ˆP end return P 19 Figure 11: Representation of the output reachable set for a single layer of the EPNM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Algorithm 13: Merge Sets (MS) Input: P = {V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , Vq}, Vi ∈ Roi×n, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , q} /* set of polytopes’ vertices */ d ∈ N /* d is the size of each group of sets that will be merged / Function MS(P, d): let A be an empty set for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , � |P| d � } do let B[1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' n] be an array for j ∈ {(i − 1) × d + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , min(i × d, |P|)} do if B = ∅ then B ← V[j] else B ← (B|V[j]) end A ← A ∪ B end return A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='1 Identification of adjacent vertices Let P = {�o i=1 λivi | �o i=1 λi = 1, λi ≥ 0 e vi ∈ V, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o}} be a convex V-polytope de- fined as the convex combination of its vertices, where V = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , vo} is the set of vertices of a polyhedron P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Given a vector c ∈ Rn and δ = max{cT x | x ∈ P}, we have that H = {x | cT x = δ} is a supporting hyperplane of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' F is a face of P if F = P or F = P ∩ H for some supporting hyperplane H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In other words, F is a face of P if, and only if, F is the set of optimal solutions for max{cT x | x ∈ P} for a given c ∈ Rn [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' vi e vj are adjacent vertices if there is a vector c ∈ Rn such that cT vi = cT vj = 20 2 V 1 V4 V9 0 V5 1 2 2 1 0 1 2 1max{cT x | x ∈ P} > cT vk, ∀k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o}, k ̸= i and k ̸= j [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Notice that the face F is an edge in the particular case stated by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We propose that: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Two extreme point vi, vj ∈ V are adjacent if, and only if, it is not possible to compute the median point, ¯v = (vi + vj)/2, as a convex combination of the vertices in V \\ {vi} and V \\ {vj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Given two vertices vi and vj, we have two possibilities regarding their adjacency: 1) they are adjacent, or else 2) they are not adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the first case, as vi and vj are adjacent, we have by definition that there is a hyperplane Ha = {x | cT x = d} that contains both vi and vj, such that Ha ∩ P = max{cT x | x ∈ P} > cT vk, ∀k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o}, k ̸= i and k ̸= j for a given c ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Therefore, all the remaining extreme points from P, except from vi and vj, are in the open half-space Hb = {x | cT x < d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Consequently, the median point of vi and vj, denoted by ¯v = (vi + vj)/2, can not be expressed as a convex combination of vk, for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} \\ {i} or k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o} \\ {j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Considering that vi and vj are not adjacent, let Pi and Pj be two polytopes given by the convex combination of the extreme points V \\ {vi} and V \\ {vj}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Hence, there are two distinct possibilities: ¯v ∈ Pj or ¯v ̸∈ Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the first case, as ¯v ∈ Pj, then we can compute this point as a convex combination of the vertices in V \\ {vj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the second case, as ¯v ̸∈ Pj and ¯v ∈ P, then ¯v ∈ P \\ Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' However, since the vertices that are adjacent to vj are in V \\ {vi, vj}, then P \\ Pj ⊆ Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' From this fact it follows that ¯v ∈ Pi and, consequently, ¯v can be expressed as a convex combination of the vertices V \\ {vi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='2 ReLU convexity inside an orthant Let f : Rn → Rn be a function that denotes the ReLU mapping, defined by the Equation (15): f(x)i = max(0, xi) (15) Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Given x, y ∈ Rn, if sup{γ | γ = sign(x)i − sign(y)i, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n}} ≤ 1, or, in other words, if x and y belong to the same orthant, we have that f(x + y) = f(x) + f(y) and that f(αx) = αf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Denoting each orthant of Rn as Oj, ∀j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , 2n}, we can rewrite the ReLU mapping, given by f, as fj : Oj → Rn, such that: fj(x) = Λjx (16) where Λj ∈ Rn×n is the matrix in which the elements of the principal diagonal associated with a negative component of x ∈ Oj are equal 0, while the remaining elements are equal 1 (λk,l for k = l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The elements off the principal diagonal are all equal to zero (λk,l = 0 for all k ̸= l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Therefore, fj preserves convexity since it consists of a linear mapping that satisfies both properties stated in the proposition, the addition property: fj(x + y) = Λj(x + y) = Λjx + Λjy = fj(x) + fj(y) 21 and the product by a scalar: fj(αx) = Λj(αx) = αΛjx = αfj(x) where α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Hence, as fj satisfies the linearity conditions within each orthant Oj and the linear mapping preserves convexity, it follows that: fj(θx + (1 − θ)y) = Λj(θx + (1 − θ)y) = θΛjx + (1 − θ)Λjy = θfj(x) + (1 − θ)fj(y) for all x, y ∈ Oj e θ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Therefore, the ReLU mapping preserves the convexity inside a given orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='3 V-polytope and half-space intersection Let P = {�o i=1 λivi | �o i=1 λi = 1, λi ≥ 0 and vi ∈ V, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o}} be a closed convex poly- hedron defined in terms of the convex combination of its vertices (or extreme points), where V = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , vn} is the set of the vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Such a polyhedron can also be defined as a set of inequalities P = {x | Cx ≤ d}, such that C ∈ Rm×n and d ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' (a) P ∩ H = P (b) P ∩ H ⊂ P (c) P ∩ H = ∅ Figure 12: Representation of all the three different possible cases with respect to the intersection of P with a half-space H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' There are three different possible cases regarding the intersection of P with a half-space H = {x | aT x ≤ b}, where a ∈ Rn and b ∈ R (Figure 12 presents a visual representation for each case): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' P ∩ H = P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' P ∩ H ⊂ P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' P ∩ H = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The first and the third cases are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the first case, we have that all the extreme points of P are in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the third case none of the extreme points of P belongs to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 22 H 2 1 V 3 P v 4 v 6 V 5H 2 3 P 4 V 6 v 5H 2 3 p V 7 4 v 6 5For the second case, a new face is generated for P, given by Ps = P ∩ Hs, where Hs = {x | aT x = b} such that a ∈ Rn and b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Note that Hs is the supporting hyperplane of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Therefore, those vertices of P that are not in H, are not extreme points of P ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Thus, it is necessary to find those extreme points of the new polytope Ph = P ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 13 presents a visualization of the elements previously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 13: Representation of the elements of interest from the intersection between P and H for the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The supporting hyperplane Hs of H is presented in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In green we have the intersection between Hs and P, and in red the result of the intersection between P with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Notice that, for this example, Vh = {v1, v2, v3, v4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We denote by Vh the subset of vertices of P that belong to H, given by Vh = V ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Let E = {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , Ep} be the set of edges of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Ei denotes the set of points that belong to the i-th edge of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Finally, we denote by Vp the set of vertices from Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Observe that Vh ⊆ Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Let c be a non-zero vector and δ = max{cT x | Cx ≤ d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The affine hyperplane Ha = {x | cT x = δ} is a supporting hyperplane of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' A subset F of P is called a face of P if F = P or F = P ∩ Ha [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' F is a face of P if, and only if, F is not empty and F = {x ∈ P | C′x = d′}, for a subsystem C′x ≤ d′ of Cx ≤ d [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Considering that P ∩ H ⊂ P and given that Vp is the set of extreme points of Ph, if v ∈ Vp and v ̸∈ Vh then v ∈ Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' By definition, an extreme point of a polytope is a 0-dimensional face, thus: F = {x | C′x = d′} (17) where C′x ≤ d′ is a subsystem of Cx ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Remember that P = {x | Cx ≤ d} and that H = {x | aT x ≤ b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Furthermore, since a vertex v is the particular case of a face given by the intersection of n hyperplanes, for a n-dimensional space, then: v = {x | C′′x = d′′} (18) for some C′′ ∈ Rn×n and d′′ ∈ Rn, such that det(C′′) ̸= 0 and v ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' However, for x ∈ H \\ Hs, no new solution for the subsystem C′′x = d′′ subsystem of: � C aT � x ≤ �d b � (19) 23 H 2 H P V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' S 3 P P 4 V 6 v 5such that det(C′′) ̸= 0 and v ̸= ∅, is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Put in other words, as no constraint was placed in H \\ Hs, no new vertex was generated in P ∩ (H \\ Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Consequently, those vertices generated by the intersection with H, in case they exist, must belong to Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Hence, if v ∈ Vp and v ̸∈ Vh, then v ∈ Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Considering that P ∩ H ⊂ P, if Vp is the set of extreme points of Ph, then Vp \\ Vh = E ∩ Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As P ∩ H ⊂ P, then Ps is a face of Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Furthermore, if F is a face of Ps, then F is a face of Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Therefore, the 0-dimensional faces of Ps (vertices) are also faces of Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Recall that a vertex is defined by the intersection of n hyperplanes, for a n-dimensional space, that P = {x | Cx ≤ d} and that H = {x | aT x ≤ b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Therewith, there are two possible cases for the vertices of Ps: v is given by C′x = d′, where C′x ≤ d′ is a subsystem of Cx ≤ d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' v is given by C′′x = d′′, where: C′′ = �C′ aT � , d′′ = �d′ b � (20) For the first case, we trivially observe that v ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the second case, as C′′ ∈ Rn×n then C′x = d′ represents the intersection of n − 1 hyperplanes, resulting in an edge of P, given by Ei = {x | C′x = d′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Hence, for the case where P ∩ H ⊂ P, if Vp is the set of vertices of Ph, then Vp \\ Vh = E ∩ Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='4 Origin regarding polytope intersection As presented in the previous section, considering that P is a closed polytope defined by the convex combination of its vertices, the intersection between P and a half-space H is given by the convex combination of the vertices from P in the intersection with H, along with the vertices obtained by the intersection of the supporting hyperplane of H with the edges of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Consequently, the intersection between P and Oj, where Oj represents one of the 2n orthants in a n-dimensional space, can be rewrite as P ∩ H1 ∩ · · · ∩ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Note that Oj = H1 ∩ · · · ∩ Hn for suitable half-spaces H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' By induction, it can be shown that the vertices of P ∩ Oj consist of the union of vertices of P that also belong to Oj with the vertices given by the intersection of the edges of P with the supporting hyperplanes of Oj (denoted as Hi, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We define Oj as: Oj = {x | Φjx ≤ 0} (21) where Φj is the suitable matrix given by: Φj = � ���� φj,1,1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 0 0 φj,2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' φj,n,n � ���� (22) such that φj,i,i ∈ {−1, 1}, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We can see that the equation system Φjx = 0 has the origin as its sole solution for any orthant j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Therefore, the unique extreme point of Oj is the origin for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , 2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 24 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' If the origin belongs to P, then it is an extreme point of P ∩ Oj, for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , 2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Let P be a convex closed polytope and Oj a convex cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' It is known that the intersection between P and Oj is a closed convex polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We have that the vertices from P that belong to Oj are also vertices of P ∩ Oj, denoted by VOj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Also, we have that P ∩ Oj = P ∩ H1 ∩ · · · ∩ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Based on Proposition 4, we have by induction that the intersection of the edges of P with the supporting hyperplanes of Oj are also vertices of P ∩ Oj, denoted by Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the origin, which is the single vertex of Oj, there are two possible cases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' the origin is not in P: this is the trivial case in which the origin can not be a vertex of P ∩ Oj, as it is not in P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' the origin is in P: in this case, by contradiction we suppose that the origin is not a vertex of P ∩ Oj, denoted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Then, there must exist two points v, u ∈ P ∩ Oj, such that: 0 = λv + (1 − λ)u (23) given that 0 < λ < 1, since 0 ̸= v and 0 ̸= u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As the origin is given by 0 = (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , 0, 0), there must exist a solution for the equation 0 = λvi + (1 − λ)ui for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , n}, such that: vi = (λ − 1)ui λ (24) Since (λ−1) λ < 0, the sign of vi and ui must by different if vi ̸= 0 and ui ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' However, inside a given orthant there must not exist a point with components that have opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Hence, by contradiction, if the origin is in P, it must be a vertex of P ∩ Oj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='5 Correctness of APNM algorithm Let F : Rn → Rm denote a neural network mapping, where L is the number of layers, the weights of the layer l are given by Wl ∈ R|xl|×|xl−1| and the biases by θl ∈ R|xl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Thereby, F(x) = (FL ◦ FL−1 ◦ · · · ◦ F1)(x) where Fl(x) = ReLU(Wlx + θl) is the mapping for each layer l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , L − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The only difference for the last layer is the activation (usually sigmoid for binary problems, or softmax for multi-class problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For a given layer l, we have that the set Xl denotes the inputs that we aim to map regarding Fl, such that Xl = {�o i=1 λivi | �o i=1 λi ∧ λi ≥ 0 ∧ vi ∈ Vl, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , o}} and Vl is the set of vertices of Xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Associated with Xl, there is Yl = {Fl(xl) | xl ∈ Xl}, which corresponds to the output set for layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Finally, let �Fl denote the output mapping for a given layer by the Algorithm 6, given by: �Fl(Vl, Wl, θl) = RP � ReLU � II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))) �� (25) where AM corresponds the affine map given by Algorithm 1, EI is the edge identification given by the Algorithm 2, II is the intersection identification defined by Algorithm 3, ReLU is given by Algorithm 4, and RP stands for removing non-vertices defined by Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We denote the convex hull mapping for a given set of vertices by CH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 25 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Given a closed convex polytope Xl as input set and Vl as the set of its vertices, then it implies that Yl ⊆ CH( �Fl(Vl, Wl, θl)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In other words, every output of a layer l associated with an input in Xl is in the convex hull of �Fl(Vl, Wl, θl) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The mapping of a layer l from F is composed of an affine map (Wlxl + θl) and a non-linear map (ReLU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Hence, since Xl is a closed convex polytope and Vl its vertices, the affine map of Xl is given by the convex hull of the affine map of its vertices, as computed by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As previously presented, the ReLU map is non-linear and therefore does not necessarily preserve the convexity of a given input set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' However, as established by Proposition 2, the ReLU mapping preserves the convexity of a convex set inside a given orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Therefore, we divide the resulting set of the affine map in such a way that each partition is inside a single orthant, so that we can apply the ReLU map to the set � Xl = CH(AM(V, Wl, θl)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As assured by Proposition 4, the intersection of a given orthant Oj with the polytope � Xl consists of the convex hull of the union of the vertices of � Xl that are in Oj, with the vertices from the intersection of the edges of � Xl with the supporting hyperplanes that define Oj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Note that Oj represents a given orthant, for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , 2|xl|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Thus, we first need to compute the edges of � Xl, calculated by EI(AM(V, Wl, θl)), as stated by Proposition 1, followed by the determination of the intersection of these edges with the supporting hyperplanes of Oj, given by II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))) and computed by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Therewith, we have that � Xl was divided in such a way that each partition is inside a single orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Then, the ReLU mapping is applied to the vertices of each partition of � Xl, resulting in ReLU(II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl)))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proposition 2 assures that the ReLU mapping preserves the convexity inside a single orthant, which allows its previous application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Note that all the vertices will be in the non-negative orthant after applying the ReLU mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Finally, we compute the convex hull of all the output sets of the ReLU mapping (at most 2|xl|) by removing those points that are not vertices of CH(ReLU(II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This final step is computed by Algorithm 5 (RP) as stated in Equation (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Consequently, CH( �Fl(Vl, Wl, θl)) is the convex hull of the exact map Yl of Xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We denote by �F the composition of �Fl for all layers of the neural network, except the last one, where the non-linear map is not applied, given by �F(V, W, θ) = ( �FL ◦ �FL−1 ◦ · · · ◦ �F1)(V, W, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Furthermore, we define Y = {F(x) | x ∈ X} as the exact output set of the network, regarding the closed convex input polytope X and its corresponding set of vertices V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Given a closed convex polytope X as the input set and V, the set of its vertices, then we have that Y ⊆ CH( �F(V, W, θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Put in different terms, each output of the network associated with an input in X is in �F(V, W, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As established by Proposition 6, Yl ⊆ CH( �Fl(Vl, Wl, θl)) for a given layer l of the neural network F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Then, for l = 1, it follows that: Y1 ⊆ CH( �F1(V1, W1, θ1)) (26) where V1 is the set of vertices from X1 and X1 = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As the output of the first layer is the input of the second one, we have that X2 = CH( �F1(V1, W1, θ1)) and that V2 = �F1(V1, W1, θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For l = 2: Y2 ⊆ CH( �F2(V2, W2, θ2)) (27) 26 Then, by replacing V2 with the output of layer 1, results in: Y2 ⊆ CH( �F2( �F1(V1, W1, θ1), W2, θ2)) (28) Now, for layers k and k + 1, it follows by induction that: Yk+1 ⊆ CH( �Fk+1( �Fk(Vk, Wk, θk), Wk+1, θk+1)) (29) Consequently, the mapping ˆF in fact computes an over-approximation for the exact output set Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='6 Correctness of EPNM algorithm The set Yl denotes the exact output associated with the input set Xl, regarding the layer l of the neural network F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The mapping of a given layer l implemented by Algorithm 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' denoted here by �El,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' is stated as: �El,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='k(Vl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Wl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' θl) = ReLU � SP � OS � II(AM(Vl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Wl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' θl),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' EI(AM(Vl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Wl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' θl))) �� k � (30) where AM is the affine map computed by Algorithm 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' EI is the edge identification implemented by Algorithm 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' II is the intersection identification given by Algorithm 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' ReLU is computed by Algorithm 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' OS is implemented by Algorithm 7 to verify if the origin is in the polytope,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' and finally SP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' which separates the vertices in orthants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' is computed by Algorithm 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Notice that k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , K} is the index that represents each �El(Vl, Wl, θl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We denote the convex hull mapping for a given input set of vertices by CH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Given a convex closed polytope Xl as input set and Vl, the set of its vertices, we have that Yl = �K k=1 CH( �El,k(Vl, Wl, θl)), where K is the number of sets that comprise �El(Vl, Wl, θl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Put another way, the set of outputs of the layer l, resulting from all inputs in Xl, consists of the union of the convex hull of each set �El,k(Vl, Wl, θl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As presented previously, the mapping of a given layer l of the neural network F is a com- position of two different functions: one affine mapping (Wlxl + θl) with one non-linear mapping (ReLU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As Xl is a closed convex polytope, the affine mapping is obtained by the convex hull of the affine map of each of its vertices, implemented by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' ReLU does not necessarily preserve the convexity of a given input set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' However, as shown by Proposition 2, the ReLU mapping preserves the convexity of an input set if the input is inside a single orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Note that the affine mapping of the input set is computed by AM(Vl, Wl, θl), where � Xl = CH(AM(Vl, Wl, θl)) denotes the application of the affine mapping in Xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Therefore, it is necessary to split � Xl in such a way that each partition lies inside a single orthant, so that it becomes possible to apply the ReLU mapping to the vertices of � Xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As assured by Proposition 4, the intersection of an orthant Oj with a polytope � Xl consists of the union of the vertices of � Xl that are in Oj, with the vertices in the intersection of the edges of � Xl with the supporting hyperplanes of Oj and the origin, if the latter lies inside � Xl, according to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Thus, we firstly compute the edges of � Xl, a step denoted by EI(AM(Vl, Wl, θl)) and computed by Algorithm 2, as stated by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Then, the intersection of these edges with the supporting hyperplanes of orthant Oj is obtained with Algorithm 3, and finally Algorithm 7 verifies whether the origin belongs to � Xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The result of such an operation is given by: OS � II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))) � (31) 27 In the next step, we separate the vertices of � Xl in different sets, such that those vertices in the same set represent the portion of � Xl that is inside a single orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This process takes place to enable the application of the ReLU mapping, as the separation allows the non-linear mapping to be applied in each partition of � Xl while ensuring convexity, as established by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Algorithm 10 performs the partitioning operation, given by: SP � OS � II(AM(Vl, Wl, θl), EI(AM(Vl, Wl, θl))) �� (32) The result of SP is the set of sets of vertices, where the convex hull of each set represents a partition of � Xl restricted to a single orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Thus, �El(Vl, Wl, θl) denotes the result of the application of the ReLU mapping to each of these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Finally, we have that �El(Vl, Wl, θl) represents the exact mapping of Xl, as we applied both the affine and the non-linear mapping without any over- approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Consequently, Algorithm 11 in fact returns the exact output set for a given layer l of the neural network F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Finally, we denote by �E the composition of �El, for each l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' , L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' To avoid repetition, the Proposition 9 is not presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' However, it follows the same inductive process as presented for the APNM (Proposition 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Given a closed convex polytope X as input set and V, the set of its vertices, we have that Y = �K k=1 CH( �Ek(V, W, θ)), where K is the number of sets in �E(V, W, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In other terms, the set of outputs of the neural network F, associated with the input set X, is in the union of the convex hull of each set �Ek(V, W, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 5 Application In this section we present a comparative analysis between the proposed vertex-based reachability approach and representative algorithms from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This comparison is carried out by verifying one of the properties from ACAS XU [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='1 ACAS XU The Aircraft Collision Avoidance System (ACAS XU) [1] comprises a set of fully connected neural networks that aim to eliminate the possibility of collisions between two aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The system comprises 45 trained models, where each model receives as inputs five properties of the ownship and the intruder (the aircraft that is invading the space of the ownship): the distance from the ownship to the intruder (ρ), the angle to the intruder regarding the ownship heading direction (θ), the heading angle of intruder relative to ownship heading direction (ψ), the speed of the ownship (vownship), and the speed of the intruder (vintruder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We can see a representation of the inputs in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' There are two extra parameters that are not used as inputs to the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The first one is the time until loss of vertical separation (τ), whereas the second one is the previous prediction advice (aprevious).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' These τ and aprevious parameters are discretized such that for each possible combination of values for τ and aprevious, a different model is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This process resulted in a total of 45 trained models, as mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Each trained model associates an input pattern to five possible categories: clear of conflict (y0), weak left (y1), weak right (y1), strong left (y3), and strong right (y4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 15 contains a simple representation of a single the neural network classifier associated with a value for τ and aprevious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Each neural network comprises 6 hidden layers, 28 Figure 14: Representation of the ACAS XU inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The black dot represents the position of the ownship and the red dot the position of the intruder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' ρ represents the Euclidean distance between both aircrafts, θ the angle between the ownship heading direction and the vector that connects both aircraft, and ψ the angle between the ownship and the intruder heading direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 15: Representation of the inputs and the outputs of a single neural network from ACAS XU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' On the left of the neural network we present each of the five expected inputs: ρ, θ, ψ, vown and vint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The neural network is defined according to a given τ and a aprev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' To the right of the neural network we have the expected probability for each output class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' CC stands for clear of conflict, WL for weak left, WR for weak right, SL for strong left, and SR for strong right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' containing 50 neurons each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For further details on the training and prediction process of these models, please refer to [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Based on the trained models of the ACAS XU, ten desired properties have been proposed so that this system works correctly according to its crafted design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In this paper, to compare with existing verification approaches, we verified Property 1, formally stated as: Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The conditions are established as follows: input constraint: ρ ≥ 55947, 691, vownship ≥ 1145 and vintruder ≤ 60;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' output constraint: y0 ≤ 1500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' where y0 is the output associated with the clear of conflict output class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 29 intruder ownship Intruder p 0 OwnshipCC p WL Neural network WR (t,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='. prev SL own SR V intRecall from the problem statement that the reachability analysis aims to verify a reachable set R, obtained from X, such that R ∩ ¬Y = ∅ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Y denotes the expected output for the inputs in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the Property 1, we have that: X = {(ρ, θ, φ, vownship, vintruder) | 55947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='691 ≤ ρ ≤ 60760, −π ≤ θ ≤ π, −π ≤ φ ≤ π, 1145 ≤ vownship ≤ 1200, 0 ≤ vintruder ≤ 60} (33) and that: Y = {(y0, y1, y2, y3, y4) | y0 ≤ 1500} (34) Property 1 is the only one presented in this paper because only this property is evaluated by means of comparison with other formal verification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' However, all of the remaining properties are formally described in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='2 Experimental description We present in this section the procedures of our experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The experimental results are divided into two main parts: the first part aiming to validate and compare the results with algorithms from the literature, and the second part to evaluate the features of our approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the first part, we conducted the experiments by: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Generate the vertices of the input polytope, based on the input constraint imposed by Property 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Set a timeout of 24 hours for each algorithm verification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Compute the output reachable set and the verification status for each reachability algorithm (Exact polytope network mapping (EPNM), MaxSens [12], Ai2 [13] and ExactReach [11]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Compute counterexamples and the verification status for the search algorithm (Reluplex [10], Duality [16], MIPVerify [15] and NSVerify [14]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Compare results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For the second experiment, which aims to analyze the parallelism behavior of the algorithm, the experimental setup consists of: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Generate the vertices of the input polytope, based on the input constraint imposed by Property 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Set the number of parallel processes, denoted by p, such that p ∈ {1, 4, 8, 12, 16, 20, 24, 28, 32};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Verify Property 1 for each p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='3 Hardware and software specification For comparative matters, we provide the specification for both hardware resources and software language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The comparative experiment previously presented was performed in an Intel Xeon CPU E5-2630 V4 of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='20GHz, with 40 available CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Those algorithms from the literature and the algorithms proposed in this work were developed in Julia language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The implementation of the algorithms from the literature were available on [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The implementation of our algorithms is available on [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='4 Comparative results The validation and comparative results of the verification of ACAS Xu models for Property 1 are presented in two different perspectives: firstly among those verification procedures that follow reachability approaches, then comparing with approaches that make use of different techniques (search or optimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Finally, we present some useful features of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='1 EPNM versus reachability approaches By comparing the EPNM approach with those verification algorithms that follow a reachability approach, as presented in Table 1, it can be seen that the proposed exact approach verified most of the neural networks within the stipulated timeout time (43 out of 45 neural networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As we can see, none of the other existing exact approaches were able to verify a single model within a day of execution (24 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Notice also that the approximate approaches (MaxSens and Ai2) finished their execution, though, due to their over-approximation, these procedures did not estimate the correct status well (which is acceptable, as these approaches are sound but not complete).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Table 1: Comparative results between the proposed approach (EPNM) regarding existing reacha- bility approaches from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' These results refer to the verification of Property 1 of ACAS XU models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proposed Approaches from the Literature Status EPNM ExactReach MaxSens Ai2 holds 43 violated 45 45 timeout 2 45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='2 EPNM versus search and optimization approaches In comparison to optimization and search approaches, the proposed EPNM approach also reached interesting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Compared to Reluplex results from the literature, EPNM could verify more neural networks within the specified timeout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' However, by executing Reluplex in the same hardware conditions of EPNM, the verification was not completed within 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The same occurred for the NSVerify procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 1Results extracted from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 2Results from the authors for a 24-hour execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 31 Table 2: Comparative results between the proposed approach regarding existing search and opti- mization approaches from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' These results refer to the verification of Property 1 of ACAS XU models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Proposed Approaches from the Literature Status EPNM Reluplex1 Reluplex (24 hours)2 Duality NSVerify holds 43 41 violated timeout 2 4 45 45 unknown 45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='3 Parallel Computation Due to the characteristics of the proposed approaches, their implementation allows the paral- lelization in a procedural level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We chose to implement the parallelization in two of the procedures (EI and II), because the remaining algorithms did not respond well due to the tradeoff between the overhead and the speed-up of the parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The first one is the edge identification (EI) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For this procedure, we created a pool for the execution with the size equal to the number of available threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The pool guarantees that, after each thread ends its execution, a new thread is started and takes the empty space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' For each vertex, a new thread was initiated, which verified if the adjacency property holds for the current and each of the remaining vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As sharing memory was not necessary, because each thread has its own adjacency list, no synchronization approach was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' By the end of the execution, the algorithm concatenated the adjacency list calculated for each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The intersection identification (II) was the second parallelized procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Following the same idea, a new thread was created for each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' After the initialization is completed, the procedure identifies those intersection points between the current and each of its adjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In this case, similarly to the previous one, no synchronization was necessary, as each thread carried its own list of intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' At the end of the pool execution, the procured concatenated those intersection points associated with each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The results of the second part of the experiments are depicted in Figure 16a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As can be seen in this figure, as the number of available threads for the execution increases, the running time decreases significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This characteristic of the algorithm can be explored for huge problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 16b presents the speedup behavior for the algorithm EPNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As expected, the algorithm indeed reduce the runtime as there is an increment on the available threads, though the difference between the real and the ideal curve indicates that the parallel processes are not ideally balanced (there are threads waiting for some execution to end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='4 Complexity behavior of the proposed approaches Our experiments showed that, differently from the expected, the algorithm EPNM has a shorter running time in comparison to APNM, for the ACAS XU model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This behavior can be explained considering the total number of vertices that are processed in both algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 17 presents the behavior of both, the total number of vertices and sets processes at each layer of a single neural network from ACAS XU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 17a shows that, from the same input set, the algorithm APNM generates the greater set of vertices for representing its approximation of the output set compared to both, EPNM and PAPNM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=', 313286 vertices at the third layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 17b depicts the increment on the number 32 (a) The graph illustrates the execution speed-up obtained from parallelizing the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As the number of available threads increases, there is a sig- nificant reduction in the running time (du- ration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' (b) Comparison between the actual speed up of the EPNM and the ideal speed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 16: Illustrative visualization of the EPNM parallelization behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' We present both, the runtime and the speed up for the algorithm EPNM execution on a single ACAS Xu model for the property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Table 3: Results on the running time of EPNM, APNM and PAPNM with d = 2 and d = 4 for the first three layers of a single neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' (a) Behavior of the total number of vertices pro- cessed for each layer of a neural network by APNM, EPNM and PAPNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Layer APNM PAPNM PAPNM EPNM (d = 2) (d = 4) 1 32 32 32 32 2 323 580 580 580 3 313286 85404 21977 2502 (b) Behavior of the total number of sets processed for each layer of a neural network by APNM, EPNM and PAPNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Layer APNM PAPNM PAPNM EPNM (d = 2) (d = 4) 1 1 1 1 1 2 1 11 21 42 3 1 78 117 149 of sets for EPNM and PAPNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Table 3 reports the data used to create Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Both results lead to a lower average number of vertices across each of the sets for EPNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' On the other hand, the opposite occurs to APNM which has a single set with a strongly increasing number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Table 4 shows the average number of vertices per set for each algorithm at each layer of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' This means that the merging process (performed partially by PAPNM and completely by APNM) induces a simplification on the total number of sets, though as a side effect it significantly increases the total number of vertices to be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The number of vertices within a set is directly related to the running time for each algorithm because the computational complexity of each of the procedures that comprise APNM, PAPNM and EPNM is directly related to the total of vertices processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 33 Runtimexthreads 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='00×104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='50×104 Runtime (s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='00×104 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='00×103 10 15 20 25 30 threadsSpeedupxthreads 32 Real Ideal 28 24 20 speedup 16 12 8 4 4 8 12 16 20 24 28 32 threads(a) Total of number vertices processed at each layer of a single neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The y-axis is in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The APNM algorithm presents a significantly higher in- crement in the number of vertices after layer 3, in comparison to PAPNM and EPNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Both PAPNM and EPNM execu- tions have the same value at layer 2, as ex- pected, though EPNM has a lower number of vertices at layer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' (b) Total number of sets generated at each layer processed by each of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' As expected, APNM keeps 1 set at the end of every layer execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The EPNM has the greatest increment on the number of sets, which is an expected behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Figure 17: Illustration of the complexity behavior of APNM, EPNM and PAPNM with d = 2 and d = 4 for the first 3 layers of a single neural network from ACAS XU model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Table 4: Average of the total number of vertices within each set for each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The results are presented layer by layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Layer APNM PAPNM PAPNM EPNM (d = 2) (d = 4) 1 32 32 32 32 2 323 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='7 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='8 3 313286 1094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='9 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='8 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='8 6 Conclusion In this work, we proposed two vertex-based reachability algorithms for formal verification of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' These algorithms compute a reachable output set, which may consist of a set of polyhedral sets, for a given input polyhedral set, satisfying different properties: the first one (APNM) computes an approximation for the output reachable set, while the second one (EPNM) computes the exact output reachable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Supported by formal demonstrations of correctness, the proposed algorithms were shown to correctly verify properties of neural networks with the ReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' More specifically, it was shown that APNM yields an overestimation of the output reachable set, while EPNM computes 34 VerticesxLayer EPNM PAPNM (d=2) PAPNM (d=4) 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0 APNM 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='5 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0 Vertices 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='5 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='5 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='0 10l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='5 1 2 m LayerSets x Layer 150 EPNM PAPNM (d=2) 135 PAPNM (d=4) APNM 120 105 90 Sets 75 60 45 30 15 0 2 m Layerthe exact reachable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Our proposal was applied to a benchmark problem for neural network verification and compared to some of the algorithms previously proposed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' The results showed that among the verification algorithms that make use of reachability analysis, the presented EPNM approach concluded most of the verifications (43 out of 45 neural networks), differently from the ExactReach, which is another exact approach from the literature that could not verify any neural network within the specified timeout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Compared to those algorithms that are not complete, despite the fact that these approaches were able to finish their executions, the expected output was not reached in any case (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' In comparison to those methods that make use of optimization and search strategies, the result reported in the literature for Reluplex surpassed ours in terms of running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' However, under the same hardware conditions, our approach was able to overcome their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Finally, we argue that the algorithms proposed in this paper are strong candidates for the ver- ification of neural networks with ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Additionally, the running time can be reduced drastically by using multiple processing cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Future work includes the investigation of a different construction for the search approach in EPNM and of a third approach that can be designed to improve performance by using heuristics for the reduction of sets and vertices during the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Acknowledgment This work was funded in part by Funda¸c˜ao de Amparo `a Pesquisa e Inova¸c˜ao do Estado de Santa Catarina (FAPESC) under grant 2021TR2265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Julian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Kochenderfer, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Owen, “Deep neural network compression for aircraft collision avoidance systems,” Journal of Guidance, Control, and Dynamics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 42, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 598–608, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Szegedy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Zaremba, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Sutskever, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Bruna, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Erhan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Goodfellow, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Fergus, “Intriguing properties of neural networks,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' abs/1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='6199, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [3] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Shlens, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Szegedy, “Explaining and harnessing adversarial examples,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' abs/1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='6572, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Madry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Makelov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Schmidt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Tsipras, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Vladu, “Towards deep learning mod- els resistant to adversarial attacks,” in International Conference on Learning Representations, ICLR 2018, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Tram`er, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Kurakin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Papernot, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Goodfellow, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Boneh, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' McDaniel, “Ensemble adversarial training: Attacks and defenses,” in International Conference on Learning Represen- tations, ICLR 2017, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Song, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Nowozin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Ermon, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Kushman, “Pixeldefend: Leveraging generative models to understand and defend against adversarial examples,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' abs/1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='10766, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 35 [7] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Grosse, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Manoharan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Papernot, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Backes, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' McDaniel, “On the (statistical) detection of adversarial examples,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' abs/1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='06280, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Metzen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Genewein, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Fischer, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Bischoff, “On detecting adversarial perturba- tions,” in Proceedings of 5th International Conference on Learning Representations (ICLR), 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Lu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Issaranon, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Forsyth, “Safetynet: Detecting and rejecting adversarial examples robustly,” in Proceedings of the IEEE International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 446– 454, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [10] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Katz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Barrett, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Dill, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Julian, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Kochenderfer, “Reluplex: An efficient SMT solver for verifying deep neural networks,” in International Conference on Computer Aided Verification, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 97–117, Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Xiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Tran, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Johnson, “Reachable set computation and safety verification for neural networks with ReLU activations,” arXiv preprint arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='08163, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [12] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Xiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Tran, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Johnson, “Output reachable set estimation and verification for multilayer neural networks,” IEEE Transactions on Neural Networks and Learning Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 5777–5783, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Gehr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Mirman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Drachsler-Cohen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Tsankov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Chaudhuri, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Vechev, “Ai2: Safety and robustness certification of neural networks with abstract interpretation,” in IEEE Symposium on Security and Privacy (SP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3–18, IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Lomuscio and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Maganti, “An approach to reachability analysis for feed-forward relu neural networks,” arXiv preprint arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='07351, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [15] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Tjeng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Xiao, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Tedrake, “Evaluating robustness of neural networks with mixed integer programming,” arXiv preprint arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='07356, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [16] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Dvijotham, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Stanforth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Gowal, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Mann, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Kohli, “A dual approach to scalable verification of deep networks,” in the Conference on Uncertainty in Artificial Intelligence (UAI), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [17] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Wong and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Kolter, “Provable defenses against adversarial examples via the convex outer ad- versarial polytope,” in International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 5286–5295, PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [18] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Kwiatkowska, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Wang, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Wu, “Safety verification of deep neural networks,” in International Conference on Computer Aided Verification, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 3–29, Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' McMullen and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Schulte, Abstract regular polytopes, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Cambridge University Press, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [20] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Emiris, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Fisikopoulos, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' G¨artner, “Efficient edge-skeleton computation for polytopes defined by oracles,” Journal of Symbolic Computation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 73, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 139–152, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Schrijver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=', Combinatorial Optimization: Polyhedra and Efficiency, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Springer, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Schrijver, Theory of Linear and Integer Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' John Wiley & Sons, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 36 [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Arnon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Lazarus, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Strong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Barrett, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Kochenderfer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=', “NeuralVerifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='jl,” 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Zago, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Camponogara, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' Antonelo, “vertexBasedRechabilityAnalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content='jl,” 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} +page_content=' 37' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INFLT4oBgHgl3EQfIy9R/content/2301.12001v1.pdf'} diff --git a/K9E1T4oBgHgl3EQfGgPl/content/tmp_files/2301.02916v1.pdf.txt b/K9E1T4oBgHgl3EQfGgPl/content/tmp_files/2301.02916v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a262307c3f6cf1f930ca6aa722a3dc024908cf37 --- /dev/null +++ b/K9E1T4oBgHgl3EQfGgPl/content/tmp_files/2301.02916v1.pdf.txt @@ -0,0 +1,1999 @@ +Unsupervised ensemble-based phenotyping helps +enhance the discoverability of genes related to heart +morphology +Rodrigo Bonazzola1,2, Enzo Ferrante3, Nishant Ravikumar1,2, Yan Xia1,2, Bernard +Keavney6,7, Sven Plein2, Tanveer Syeda-Mahmood4, and Alejandro F Frangi1,2,5,* +1Centre for Computational Imaging and Simulation Technologies in Biomedicine (CISTIB), School of Computing and +School of Medicine, University of Leeds, Leeds, UK +2Leeds Institute of Cardiovascular and Metabolic Medicine, School of Medicine, University of Leeds, Leeds, UK +3Research Institute for Signals, Systems and Computational Intelligence, sinc(i), FICH-UNL / CONICET, Santa Fe, +Argentina +4IBM Almaden Research Center, San Jose, USA +5Medical Imaging Research Center (MIRC), University Hospital Gasthuisberg. Cardiovascular Sciences and +Electrical Engineering Departments, KU Leuven, Leuven, Belgium +6Division of Cardiovascular Sciences, Faculty of Biology, Medicine and Health, University of Manchester, +Manchester, UK +7Manchester University NHS Foundation Trust, Manchester Academic Health Science Centre, Manchester, UK +*a.frangi@leeds.ac.uk +ABSTRACT +Recent genome-wide association studies (GWAS) have been successful in identifying associations between genetic variants +and simple cardiac parameters derived from cardiac magnetic resonance (CMR) images. However, the emergence of big +databases including genetic data linked to CMR, facilitates investigation of more nuanced patterns of shape variability. Here, +we propose a new framework for gene discovery entitled Unsupervised Phenotype Ensembles (UPE). UPE builds a redundant +yet highly expressive representation by pooling a set of phenotypes learned in an unsupervised manner, using deep learning +models trained with different hyperparameters. These phenotypes are then analyzed via (GWAS), retaining only highly confident +and stable associations across the ensemble. We apply our approach to the UK Biobank database to extract left-ventricular (LV) +geometric features from image-derived three-dimensional meshes. We demonstrate that our approach greatly improves the +discoverability of genes influencing LV shape, identifying 11 loci with study-wide significance and 8 with suggestive significance. +We argue that our approach would enable more extensive discovery of gene associations with image-derived phenotypes for +other organs or image modalities. +Introduction +Genome-wide association studies (GWAS) have remarkably accelerated discoveries of associations between genomic and +complex traits [1]. In general, they analyse genetic variants (i.e the genotype) in a sample of individuals, to test their possible +association with the presence of disease or with systematic changes in measurable traits, known broadly as phenotypes in +this context. GWAS have already successfully identified genetic variants associated with a broad range of diseases and other +complex traits, such as metabolic, anthropometric or behavioural ones. These findings have improved our understanding of the +pathogenesis of disease, facilitating the development of better treatments, supporting drug discovery and assisting advances +towards precision medicine. +Large-scale epidemiological imaging studies have correlated image-derived phenotypes (IDPs) to genetic data for identifying +the genetic basis of organ structure and function in health and disease. In cardiology, GWAS have been performed on clinically +relevant quantitative indices of the left ventricle (LV), such as LV volumes, LV mass and LV ejection fraction, as the diagnosis +of patients with heart disease typically starts from quantitative analysis of this cardiac chamber [2, 3]. Although there are +discrepancies in the number of genetic loci associated with changes in LV IDPs from recently reported GWAS [2, 4], some +consistent genetic factors have been established. +These cardiac imaging genetics studies were based on traditional approaches, where handcrafted features characterising LV +IDPs were first determined, before running GWAS to find the associated genetic loci. Although these IDPs have been clinically +arXiv:2301.02916v1 [q-bio.GN] 7 Jan 2023 + +Figure 1. Flowchart of the proposed Unsupervised Phenotype Ensembles (UPE) framework. First, a graph-convolutional +autoencoder is trained and applied to our set of left-ventricular meshes (number of vertices M = 5220) to produce +low-dimensional representations of these shapes. In each layer, a representation with fewer vertices is obtained. The bottleneck +zθr of the autoencoder with hyperparameters θr is a nr +z-dimensional vector for each run r (nr +z ∈ {8,16}). The different latent +vectors obtained for each run, {zθr}R +r=1 are then tested in a GWAS component by component, for association with genetic +variants. +used to diagnose heart disease, they do not provide detailed representations of the chamber’s morphology and its variation +across the population. In this paper, we advance the view that shape features encoded in a learned latent space can provide a +more refined imaging phenotype which turns out to be more informative than traditional measurements. When associated with +genetic variation, this can provide novel insights into the genetic basis of cardiac structure and function. +The unprecedented amount of linked genetic and cardiac imaging data available within the UK Biobank (UKB) [5] facilitates +the use of unsupervised machine learning techniques to automatically learn a set of features that best describe the morphology +of the heart. The idea of using unsupervised learning to produce phenotypes that can be tested in GWAS has been previously +explored by us in a conference paper [6], and also recently utilised in [7] in order to discover genetic loci related to phenotypes +derived from retinal fundus images. +Atlas-based methods have been proposed to generate three-dimensional (3D) meshes representing cardiac anatomy from +volumetric images [8, 9]. We build on top of these works, leveraging the latest advances in graph-convolutional neural networks +[10] to learn low-dimensional representations that consider mesh topology. While standard convolutional neural networks +operate on domains with an underlying Euclidean or grid-like structure (e.g. images), geometric deep learning generalises +convolutions to non-Euclidean domains such as graphs, meshes and manifolds, taking into account their topology and irregular +structure. Previous studies employed mesh autoencoders in modelling the expression space of human face surfaces [11]. Here +2/36 + +R +runs +Input mesh +Reconstructed mesh +M×3 +: spectral graph convolution + pooling + ReLU +M×3 + fully connected layer +0r E 0 +latent representation +Z0R +nz × 1we show that such models can enable anatomical variation in cardiac structures to be learned and correlated to genetic data. +In this work, we learn compact and non-linear representations of cardiac anatomy in an unsupervised setting via convolutional +mesh autoencoders (CoMA). We hypothesise that the learned features can identify genetic loci that impact cardiac morphology +due to their ability to explain shape variability across the population. We show that such representations can indeed be used to +discover novel genetic associations via GWAS, which was not previously possible with traditional handcrafted IDPs such as +volume, mass and function indices. +A schematic overview of the proposed UPE method is presented in Fig. 1. The details of each step are outlined in the +Methods section. First, we extracted a surface mesh representation of the anatomical structures. In particular, we studied 3D +meshes representing the LV at end-diastole from CMR images of the UK Biobank database using an automatic deep-learning- +based segmentation method [12]. We then learn a low-dimensional representation of the 3D meshes which captures anatomical +variations using an encoder-decoder model. All meshes were projected onto this latent space to derive a few shape descriptors +(or latent variables) for each of them. These features were finally used in GWAS for discovering genetic variants associated +with shape patterns. Furthermore, to enhance discoverability, we adopt an ensemble-based approach: a set of phenotypes +obtained through different models trained and configured with varying network metaparameters and weight initialisations +(which induce diversity in the learned representations) are pooled together in one ensemble, yielding a redundant yet more +expressive representation than the individual latent vectors. GWAS is performed against each phenotype of the ensemble, one +at a time. A corrected Bonferroni threshold is then computed to declare the associations significant. We demonstrate that this +approach is indeed effective in discovering additional biologically relevant genetic associations. +Results +Convolutional mesh autoencoders were trained on the LV meshes at end-diastole, using a range of network metaparameters. +For comparison, we also fit a shape PCA model (see Methods section). The reconstruction performance for these models is +displayed in Supplementary Figure S6. +GWAS was performed across all latent variables, for all training runs achieving a good reconstruction performance (see +Methods section). The number of such runs was R = 36. Results obtained with nz = 8 and nz = 16 (eight and sixteen latent +variables respectively) are reported, with a total number of 384 latent variables in the pooled representation. We also note that, +despite the good reconstruction performance, shape PCA was not able to find genome-wide significant association when tested +in GWAS (see Supplementary Figure S7). +Firstly, we examined the prevalence of the significant GWAS loci found, across runs. To count the loci, we split the genome +into approximately LD-independent genomic regions [13] and computed the number of loci below the usual genome-wide +significance threshold of 5×10−8 (see details in the Methods section); Table 1 shows the results. Loci were annotated with +gene names using the web tool FUMA (Functional Mapping and Annotation) [14]. Among the candidate genes provided by +this tool, a literature review was conducted in search for evidence of an association with cardiovascular phenotypes. +Genetic findings +Loci with study-wide significance. +We observe that the five most prevalent loci have prior evidence of association to cardiac +phenotypes. PLN plays a crucial role in cardiomyocyte calcium handling by acting as a primary regulator of the SERCA protein +(sarco/endoplasmic reticulum Ca2+-ATPase), which transports calcium from the cytosol into the SR1 [15].Mutations in PLN +have a well-established relationship with dilated cardiomyopathy (DCM) [16]. In [4], PLN was found to be associated with +LVEDV and LVESV. However, [2] does not report this locus for the same phenotypes. +The locus at chromosome 2 has been reported in [2, 4, 17] and mapped to gene TTN. This gene encodes for protein titin, +which is responsible for the sarcomere assembly of the myocytes, and determines stretching, contraction and passive stiffness +of the myocardium [18]. +The association with SNP rs4767239 is likely linked to gene TBX5 (T-box transcription factor 5), which has a known role +in the development of the heart and the limbs [19]. Mutations in this gene have been associated, through familial studies, with +Holt-Oram syndrome, a developmental disorder affecting the heart and upper limbs. Notably, it has not been reported on recent +GWAS on LV phenotypes. +SNP rs35564079 is likely associated to gene NKX2.5. As it happens with TBX5, it plays a crucial role in heart development; +in particular, in the formation of the heart tube, which is a structure that will eventually give rise to the heart and great vessels. +NKX2.5 helps to determine the position of the heart in the chest and also plays a role in the development of the heart valves +and septa. Mutations in the NKX2.5 gene have been linked to several types of congenital heart defects, including atrial septal +defects and atrioventricular block [20]. It has not been reported in [2] or [4] but shows borderline significance with trabecular +fractal dimension [17]. +Interestingly, [17] reports the GOSR2 locus as significantly associated to trabecular fractal dimension at slices 3 and 4, +however previous GWAS on global LV indices have not reported this locus. More broadly in the literature on genetics of +3/36 + +cardiovascular phenotypes, it has been reported as associated to ascending aorta distensibility [21], mitral valve geometry [22] +and congenital heart disease [23]. +The association at SNP rs2245109 is likely linked to gene STRN, in chromosome 2. This gene codes for protein striatin, +which is expressed in cardiomyocytes and it has been shown to interact with other proteins implicated in the mechanism of +myocardial function [24]. Moreover, mutations in this gene have been shown to lead to DCM in dogs. [25]. +The locus near gene ATXN2 has been previously reported for LVEDV and LVSV. [4] +In addition to the loci with prior evidence discussed above, we report a number of novel genetic loci with p < pSW, which +have not been previously reported in connection with cardiac phenotypes. We note that all of these novel associations were +detected in over 25% of the runs (see Table 1). The loci were annotated based on the closest gene: CCDC91, LSP1, LGALS8 +and EN1. We note that locus EN1 is also significant for LVEDSph (see Supplementary Figure S2). +The GWAS summary statistics for the best latent variable of each of these 11 loci is displayed in Table 2. +Loci with suggestive significance. +In addition to genetic loci with p < pSW, there is a number of SNPs with pSW < p < pGW, +which we mention here by virtue of them showing up in more than 5 independent runs of the autoencoder. +We first examine the loci close to genes with some known roles in cardiac physiology: genes BAG3, WAC, LMO7, RBM20 +and CNOT7. BAG3 is a gene coding for a cellular protein that is expressed predominantly in skeletal and cardiac muscle, +which has a role in homeostasis of myocytes and in the development of heart failure [26]; also, it shows a strong association +with LVESV, as found in previous studies [2, 4] and in our own GWAS for this phenotype. The association near WAC is +likely causally linked to the WAC gene; indeed, pathogenic deletions in this gene have been shown to lead to rare genetic +disorders that produce, among other phenotypes, cardiac defects [27]. However, further investigation would be needed to assess +this hypothesis. Mutations in the LMO7 gene have been associated to Emery–Dreifuss muscular dystrophy, a disease with +cardiac manifestations. Variants in the RBM20 are associated to DCM [28]. Finally, CNOT7 codes for a protein, CCR4-NOT, +which is a subunit of a protein complex called CCR4-NOT deadenylase whose activity was found to be required to prevent +cardiomyocyte death in mice [29]. +Three additional stable loci were found near genes CENPW, FILIP1L, and OR9Q1. +Effect on LV morphology. +The effect of these loci on LV morphology was assessed by selecting the single phenotype with the strongest p-value for +the associated locus. To help characterise these latent variables, Spearman correlation coefficient between the latter and LV +handcrafted indices were computed and shown in Supplementary Table S3. These indices were LVEDV, LV sphericity index at +end-diastole (LVEDSph), LV myocardial mass (LVM), and LV mass-to-volume ratio (LVMVR=LVM/LVEDV). +We also examine the shapes of the average mesh for different ranges of quantiles for this latent variable, from 0 through 1. +This is shown in Fig. 2, along with the associated Manhattan plots, for loci PLN and TTN. For each of the remaining loci, the +morphological effect for the latent variable with the strongest association can be found in the Supplementary Material, along +with Manhattan plots and LocusZoom plots in a 1 Mb region centered at the locus. +Interestingly, we observe a very distinct effect on morphology for each of these loci. Whereas PLN variants influence +a latent variable which is mainly linked to the sphericity (Spearman r = 0.625) with a relatively small effect on LVEDV +(r = 0.434), gene TTN shows a greater correlation with the latter (r = 0.889). Consistent with this, GWAS on LVEDSph shows +no signal for TTN, but a strong one for PLN (p = 10−15, see Supplementary Figure S2), which is also in line with a previous +finding of ours [6]. +On the other hand, gene TBX5 is linked to a latent variable which, as for TTN, is mainly correlated to LVEDV with no +correlation to LVEDSph. For developmental gene NKX2.5 (see Supplementary Figure S12), we note that the associated latent +variable has an effect both in LVEDV (r = 0.865) as in LVEDSph (r = 0.372). STRN locus is linked to a subtle phenotype +controlling mitral orientation without a concomitant change in LV size (see Supplementary Figure S22). +Comparison with GWAS on traditional LV indices. +For comparison, we collected the GWAS summary statistics from previous studies on LV phenotypes, derived also from UKB +CMR images, namely: [2], [4] and [17]. We also include LVESV, LVSV and LVEF. Note, however, that the unsupervised +features studied in this work are static and were extracted using only the end-diastolic phase. +We also performed our own GWAS on traditional cardiac indices obtained using our segmentation approach (i.e. the indices +were derived using the same meshes as the unsupervised phenotypes). Note that LVEDSph has not been investigated in previous +GWAS, and is reported for the first time here, to the best of our knowledge (although a related phenotype, named "LV internal +dimensions" was studied in an early GWAS of echocardiography-derived LV traits, [30]). Details on how to compute this +phenotype can be found in the Supplementary Material. +The comparison can be seen in figure 3. For each locus in Table 2 (which all passed the Bonferroni threshold), this figure +displays the association p-value found in previous GWAS. Shades of red represent non-genome-wide significant associations, +4/36 + +region (hg37) +locus name +count +minimum p-value +chr6:117672972-118963115 +PLN +35 +1.8×10−22 +chr2:178553183-181312739 +TTN +34 +5.3×10−14 +chr12:113986709-115036602 +TBX5 +31 +5.5×10−12 +chr17:43056905-45876022 +GOSR2 +30 +8.0×10−15 +chr12:110336719-113263518 +ATXN2* +26 +1.1×10−11 +chr5:171074292-172678327 +NKX2.5 +26 +1.8×10−11 +chr13:75670143-77410555 +LMO7 +23 +6.6×10−09 +chr6:125424383-127540461 +CENPW* +23 +2.3×10−10 +chr10:110317705-112561493 +RBM20 +21 +4.3×10−09 +chr11:1213590-3665481 +LSP1* +20 +1.6×10−12 +chr8:15991660-17387876 +CNOT7 +17 +7.4×10−10 +chr10:26888684-29323236 +WAC +14 +3.1×10−09 +chr1:235819436-237555628 +LGALS8* +14 +1.2×10−10 +chr12:27799773-29651255 +CCDC91* +13 +7.6×10−13 +chr3:99373762-100592217 +FILIP1L* +12 +1.0×10−09 +chr2:118367466-121303783 +EN1* +11 +1.3×10−10 +chr10:120591353-122407323 +BAG3 +8 +8.5×10−09 +chr11:55082657-58457495 +OR9Q1 +7 +1.4×10−09 +chr2:36122006-38132712 +STRN +6 +7.0×10−11 +Table 1. Counts of GWAS hits across runs, Cℓ for each locus ℓ, which represents the number of runs for which the +corresponding locus shows at least one association with p < pGW = 5×10−8 (see details in the Methods section). Note that +the total number of runs was 36. Only regions with more than 5 counts are shown here. Genes with an asterisk were annotated +based purely on proximity to the lead variant in that region. Gene names with no asterisk have additional prior evidence of a +link to cardiac physiology. The last column represents the minimum p-value across the phenotype ensemble. Loci that surpass +the study-wide threshold pSW(= 1.5×10−10) are shown in bold letters. +locus name +chromosome +position (hg37) +lead variant +| ˆβ|±sd( ˆβ)(×10−2) +p-value +PLN +6 +118667522 +rs11153730 +7.7±0.8 +1.8×10−22 +GOSR2 +17 +45013271 +rs17608766 +9.2±1.1 +3.9×10−16 +TTN +2 +179558366 +rs2042995 +7.2±1.0 +5.3×10−14 +CCDC91* +12 +28544464 +rs3741760 +6.6±0.9 +7.6×10−13 +LSP1* +11 +1887068 +rs569550 +5.8±0.8 +1.6×10−12 +TBX5 +12 +114816548 +rs4767239 +7.3±1.1 +5.6×10−12 +ATXN2* +12 +111907431 +rs35350651 +5.4±0.8 +1.1×10−11 +NKX2.5 +5 +172670611 +rs35564079 +6.0±0.9 +1.8×10−11 +STRN +2 +37086197 +rs2245109 +5.2±0.8 +7.0×10−11 +LGALS8* +1 +236691532 +rs2853621 +5.5±0.9 +1.2×10−10 +EN1* +2 +119479427 +rs162748 +5.5±0.9 +1.3×10−10 +Table 2. 11 genetic loci that surpass the study-wide significance threshold of pSW = 1.5×10−10, along with the summary +statistics for the best phenotype for each locus. ˆβ and sd( ˆβ) are the estimated effect sizes and their standard errors, +respectively; these correspond to the inverse-rank-normalised phenotypes. +whereas shades of blue represent genome-wide significant ones, and white corresponds to the pGW threshold. The second row +represents the best p-value across all the traditional phenotypes for the loci given in the columns. +5/36 + +Figure 2. Manhattan plots for LV latent variables with best association with a) PLN and b) TTN loci. On top are shown the +average meshes corresponding to the following range of quantiles, for each latent variable: [0, 0.01], [0.095, 0.105], [0.495, +0.505], [0.895, 0.905] and [0.99, 1]. +Discussion +As exposed in the Results section, we were able to retrieve study-wide significant loci that had been found in previous GWAS +on handcrafted phenotypes (PLN, TTN, ATXN2 and GOSR2). In addition, genes with a known role in cardiac physiology +(TBX5, NKX2.5, STRN) but no prior GWAS association, were identified with the same level of significance. Four additional +loci constitute potential avenues for future research. Finally, eight additional loci were found with suggestive significance +(pSW < p < pGW and Cℓ > 5). 5 of these have a prior evidence of a link to cardiac pathways. +It is interesting to note that, for some loci, a relatively small number of runs produced a latent variable with a genome-wide +significant association to the locus: the UPE approach seems to be crucial in order to retrieve this association, as it is likely to +be missed in one individual autoencoder run. +Importantly, our approach allows to detect the milder effect on morphology of common variants near genes whose mutations +are known to have highly deleterious effects (either by study of Mendelian diseases in humans, or by studies on model +6/36 + +20 +PLN +15 +(d)0T60]- +10 + +10 +11 +12 +13 +14 +15161718 +20 +22 +chromosome +14 +12 +TTN +10 +(d)01601- +10 +11 +12 +13 +14 +15161718 +20 +22 +chromosomeFigure 3. Comparison of the −log10(p) values for the 11 study-wide significant genetic loci found in this work, with GWAS +on handcrafted cardiac indices. The top row corrsponds to the best association found for that locus across the ensemble of +phenotypes, whereas the second row corresponds to the best p-value for that locus across the previous GWAS. White color +corresponds to the genome-wide significance threshold of 5×10−8, whereas shades of red and blue correspond weaker and +stronger associations, respectively. SV denotes stroke volume. LVEDVi, LVESVi and SVi denote the indexed versions of the +phenotypes, i.e. the phenotype divided by the subject’s body surface area. +organisms). It is likely that these variants and the associated unsupervised LV features hold prognostic value, however this is +uncertain at this point, and it should be possible to assess it once UKB releases more longitudinal data on the same subjects +studied here. +As an interesting observation, we note that most of the phenotypes extracted here show a remarkable oligogenicity, i.e. +they are controlled by few genes (see Figures S8 through S22). This is in contrast to what happens with traditional indices +where there are often many associations (see Supplementary Figures S1 through S5 or [2, 4, 17]). This suggests that the UPE +approach allows to identify more optimal phenotypes for each locus, as compared to traditional handcrafted phenotyping. +In terms of gene discovery, the advantages of the ensemble-based phenotyping approach proposed here are best conveyed +by examining the association p-values of the loci found in GWAS performed against traditional handcrafted phenoytpes, +displayed in Figure 3. For example, when examining the GOSR2 locus, we found no genome-wide significant association +7/36 + +CREBRF* +SR2 +TXN2* +TOP* +LSP1* +TRN +5 +B +S +V +G +Best association (CoMA ensemble +Best association (from other GWAS) +OurGWAS__LVEDV +OurGWAS_LVEDSph +OurGWAS__LVM +AUNG2019_LVMVR +AUNG2019_LVEDV +AUNG2019__LVM +AUNG2019_LVEF +AUNG2019LVESV +PIRRUCCELLO2020__LVEDV +PIRRUCCELLO2020__LVEDVi +PIRRUCCELLO2020__LVESV +PIRRUCCELLO2020__LVESVi +PIRRUCCELLO2020LVEF +PIRRUCCELLO2020__SV +PIRRUCCELLO2020 SVi +MEYER2020__LV_fractal_dim_slice1 +MEYER2020__LV_fractal_dim_slice2 +MEYER2020__LV_fractal_dim_slice3 +MEYER2020 LV fractal dim slice4 +MEYER2020__LV_fractal_dim_slice5 +MEYER2020__LV_fractal_dim_slice6 +MEYER2020__LV_fractal_dim_slice7 +MEYER2020__LV_fractal_dim_slice8 +MEYER2020__LV_fractal_dim_slice9 +01.5 3 4.5 6 7.5 9 10.5 12 13.5 15when performing GWAS on traditional LV indices derived from the same meshes; neither have previous studies, with the +exception of [17] which investigated LV trabecular fractal dimension. Likewise, other genes, like STRN, which have prior +knowledge of being implicated in cardiac pathways, have not been reported to date in mostly healthy cohorts such as UKB. +Other examples are the loci near genes CCDC91 and LSP1, which achieve study-wide significance in our approach, however +they have been reported in no previous GWAS on LV phenotypes (i.e. all squares are coloured with red shades). This highlights +the shortcomings of traditional image-derived-phenotyping techniques when it comes to discoverability of relevant genes +In addition to improved discoverability, the UPE framework enables a better understanding of the genetic architecture of +cardiac phenotypes, even for genetic loci that were known from previous studies. Most notably, TTN was shown to be linked to +LV size, whereas PLN (which has been previously found in GWAS of LVEDV) controls a feature that jointly models changes +in LV size and sphericity. +Based on our findings we argue that, in large-scale imaging studies it is crucial, along with increasing sample size, to count +with good techniques to perform deep phenotyping that allows to boost gene discoverability in GWAS. +Conclusions +In this work, we proposed a framework for left-ventricular (LV) phenotyping based on unsupervised geometric deep learning +techniques on image-derived 3D meshes to discover genetic variations that affect LV shape through GWAS. The proposed +methodology is based on finding a latent low-dimensional representation of the CMR-derived LV 3D meshes using convolutional +mesh-autoencoders and then performing GWAS on the learned latent features. As hypothesised, this dimensionality reduction +method, using Kullback-Leibler regularisation, yielded phenotypes with statistically significant genetic associations. +The methodology of ensembling SNP associations across representations obtained through different network metaparameters, +followed by the correction in the Bonferroni threshold necessary to control for false discover rate, has proven effective in +identifying novel associations of mesh-derived phenotypes with genetic loci. In addition to previously identified loci, namely +TTN, PLN, TBX5, GOSR2 and ATXN2, we report 6 additional genetic loci that have not been discovered in recent GWAS of +LV phenotypes. Moreover, we report a number of loci which do not surpass our corrected Bonferroni threshold, however their +association remains suggestive by virtue of surpassing the usual genome-wide significance threshold of pGW = 5×10−8 in a +number of phenotypes, obtained from independently trained autoencoder networks. Some of the latter, like BAG3, RBM20 and +STRN, have been previously linked to cardiac pathways as have not been previously linked to cardiac physiology. +We argue that the proposed ensembling approach is not only useful for discovering novel associations, but also enables +a deeper understanding of the effect of previously known genes: indeed, the effect of the latent variables with the strongest +association p-values for each locus can be used as suggestive evidence of the role of that locus in LV shape. For example, we +found that TTN and PLN, which had been found in previous GWAS to correlate with LV volume, have actually a distinct effect +on LV shape. Whereas TTN shows indeed a clear effect on LV size, PLN is linked to a more complex phenotype which mainly +involves changes in LV sphericity. +More generally, these results validate our methodology to extract knowledge about the genetics driving the morphology of +organs, leveraging databases that provide linked genetic and imaging data, such as the UKB. This methodology can be used +seamlessly to study surface meshes of other organs, like the brain or the skull [31, 32]. Additionally, the algorithm proposed +here can be extended to process 3D cardiac meshes throughout the cardiac cycle to capture anatomy as well as quantitative +features related to patterns of contraction and relaxation. Future studies will explore these directions. +Methods +The proposed method is outlined in Figure 1. It starts with the extraction of 3D meshes representing the LV from CMR +images using an automatic segmentation method [12]. We then train several models with different metaparameters (network +architecture, random seeds controlling weight initialization and dataset partitioning, and relative weight of the variational loss) +to learn low-dimensional representations of the 3D meshes which capture anatomical variations using an encoder-decoder +model. All meshes are then projected to this latent space to derive a few shape descriptors (or latent variables) for each mesh. In +order to take advantage of the variability induced in the representation obtained by the metaparameters, we pooled the different +latent vectors together to obtain a richer representation. The features that constitute this pooled representation are finally used +in GWAS to discover genetic variants associated with shape patterns. +Description of the data +The proposed framework can discover novel associations between genetic variations and morphological changes in anatomical +structures. We showcase its potential in the context of cardiac images acquired within the UK Biobank (UKB) project (data +accession number 11350). The UKB is a prospective cohort study that between 2006 and 2010 recruited around half a million +volunteers across the United Kingdom, aged 40-69 years old at the time of recruitment [5]. The project collected a vast amount +8/36 + +of phenotypic information about its participants and linked them to their electronic health records (EHR). The collected data +includes, among others, genetic data from single-nucleotide polymorphism (SNP) microarrays for all the individuals, and also +CMR data for a subset of them (which comprises over 50,000 individuals at the moment of this writing, but is planned to reach +100,000). These datasets are described in [33] and [34], respectively. +CMR data +The CMR imaging protocol used to obtain the raw imaging data is described in [34]. We employed an automatic segmentation +method [8] to segment the LV in the CMR images. This method generates a set of registered 3D meshes, i.e. meshes with +the same number of vertices with consistent identical connectivity between them. There is one mesh per subject and per time +point. In this work, we only use the LV mesh at end-diastole. The LV mesh for subject i, i = 1,...,N, can then be represented as +pairs (Si,A), where Si = +� +xi1 yi1 zi1 |...|xiM yiM ziM +� +∈ RM×3 is the shape and A is the adjacency matrix of the mesh. We also +define, for convenience, the vectorised form of the shapes si = +� +xi1,yi1,zi1,...,xiM,yiM,ziM +� +∈ R3M. The adjacency matrix is +such that A jk = 1 if and only if there is an edge between vertices j and k and Ajk = 0 otherwise. The cardiac meshes also have +the property of being triangular and closed, so A jk = Akl = 1 =⇒ Ajl = 1 for all vertices i, j and k. +Genotype data +SNP microarray data is available for all the individuals in the UKB cohort. This microarray covers 801,526 genetic variants +including SNPs and short insertions and deletions. The SNP microarrays used in UKB have been described in [33]. From these +genotyped markers, an augmented set of over 90 million variants was imputed. The GWAS was performed across the latter +dataset, particularly on the autosomes (chromosomes 1 through 22). +The usual quality control steps on the genetic data were performed. This included filtering out rare variants using a +threshold for MAF of 1% (within the subcohort of 31,602 subjects), Hardy-Weinberg equilibrium p-value less than 10−5 and +low imputation information score (less than 0.3). This results in a set of 9,472,708 genetic variants. +Unsupervised representation learning for genetic discovery +Given the set of meshes representing the anatomical structure of interest (LV meshes), the pose-sensitive parameters (translation +and rotation) were removed using generalised Procrustes analysis. +¯S = 1 +N +N +∑ +i=1 +S, +(1) +¯s = 1 +N +N +∑ +i=1 +s, +(2) +C = +1 +N −1 +N +∑ +i=1 +(si − ¯s)(si − ¯s)t. +(3) +Here we propose to learn a reduced set of features that best describe cardiac shape using convolutional mesh-autoencoders +(CoMA). We will compare the proposed approach with the well-known method of principal component analysis (PCA). While +in PCA only vectorised 3D point clouds si will be provided as input (therefore ignoring the data structure and topology), +convolutional mesh-autoencoders leverage topological information about the connectivity between the vertices for learning more +powerful non-linear representations. However, both approaches can be thought of as particular cases of the encoder-decoder +paradigm. +In such a paradigm, there is a pair of encoding and decoding functions, Eθ : R3M → Rnz and Dφ : Rnz → R3M that are +parameterised by a set of learnable coefficients θ and φ, respectively. nz ∈ N is the size of the latent space, and it is usually +chosen so that nz ≪ M (hence the dimensionality reduction). +Optimal parameters θ ∗ and φ ∗ for reconstruction can be estimated by making the composite function Dφ ◦Eθ as close to +the identity function I as possible over the training set Strain ⊂ S, using some reasonable measure of reconstruction error Lrec +(examples of which are the L1 norm, the L2 norm or the mean squared error MSE) along with a regularisation term Ω, which +will account for additional constraints we want to impose on the model. We want to minimise the following function with +respect to φ and θ: +L(Strain|θ,φ) = Lrec(Strain|θ,φ)+βΩ(Strain|θ,φ). +(4) +where β ∈ R≥0 is a weighting coefficient for the regularisation term. zi := Eθ∗(Si) ∈ Rnz would then be a low-dimensional +representation of the shape Si, whereas ˆSi := +� +Dφ∗ ◦Eθ∗� +(Si) is the associated reconstructed shape. +9/36 + +Principal component analysis. +PCA is a standard linear technique for dimensionality reduction [35]. In terms of the encoder-decoder framework detailed +above, it can be obtained by requiring D and E to be linear transformations and using the L2 norm, besides imposing an +orthogonality constraint on the latent vectors [36]. +The idea is to find a basis of vectors Bnz = {vi}nz +i=1 ⊂ R3M for a fixed nz < 3M. The nz-dimensional linear subspace +generated by Bnz captures as much of the data variability as possible. It can be shown that such a basis corresponds to the nz +eigenvectors of C with the largest eigenvalues; i.e. if C = UtΛU where Λij = δijλi and λi ≥ λ j if i ≤ j, then Bnz = {Uei}nz +i=1. +δi j is the Kronecker delta, which equals 1 if i = j and 0 otherwise. +Convolutional mesh autoencoder +In an autoencoder, both the encoding and decoding functions are feed-forward neural networks. Inspired by recent works +on unsupervised geometric deep learning [11] for facial meshes, we propose constructing a convolutional mesh-autoencoder +which uses spectral convolutions [37] to learn non-linear and low-dimensional representations of cardiac mesh structures. Here +each layer of the encoder and decoder implements convolution operations parameterised by the graph Laplacian, to leverage +information about the local context of each vertex. In order to learn global features, a hierarchical approach is used where each +layer of the encoder and decoder implements downsampling and upsampling operations, respectively. Since the vertices are not +in a rectangular grid, the usual convolution, pooling and unpooling operations defined for such topology (usually employed in +image analysis) are not adequate for this task and need to be suitably adapted. Several methods have been proposed to do this +[10] which can be mainly classified in two broad groups: spatial or spectral. The approach proposed in this work belongs to the +latter category, which relies on expressing the features in the Fourier basis of the graph, as explained below. +Spectral convolutions. +The Laplace-Beltrami operator L (or, more simply, the Laplacian) of a graph with adjacency matrix +A is defined as L := D−A, where D is the degree matrix, i.e. a diagonal matrix with Dii := ∑j Aij being the number of edges +that connect to vertex i. The Fourier basis of the graph can be obtained by diagonalising the Laplace operator, L = UtΛU. +The columns of U constitute the Fourier basis, and the operation of convolution ⋆ for a graph can be defined in the following +manner: +x⋆y := U(Utx⊙Uty), +(5) +where ⊙ is the element-wise product (also known as Hadamard product), and x and y are arbitrary functions defined over the +vertices of the graph. Spectral methods rely on this definition of convolution and differ from one another in the specific filter +used. In this work, a parameterisation proposed in [37] will be used. The said method is based on the Chebyshev family of +polynomials {Ti}. The kernel gξ is defined as: +gξ(L ) = +K +∑ +i=1 +ξiTi(L ). +(6) +K is the highest degree of the polynomials considered (in this work K = 6). Chebyshev polynomials have the advantage of +being computable recursively through the relation Ti(x) = xTi−1(x)−Ti−2(x) and the base cases T1(x) = 1 and T2(x) = x. It is +also worth mentioning that the filter described by Equation 6, despite its spectral formulation, has the characteristic of being +local. +Autoencoder. +The downsampling and upsampling operations used in this study were proposed in [11] based on a surface +simplification algorithm proposed in [38]. These operations are defined before training each layer, using a single template +shape. Here we utilise the mean shape ¯S as a template. +In each layer of the encoder, the downsampling operation generates a new triangular mesh (with its corresponding new +Laplacian), such that the quadric error is minimised. The upsampling operations are created while performing the downsampling: +the coordinates of the decimated vertices with respect to the remaining vertices are stored for each layer. +Variational Autoencoder. +A Kullback-Leibler (KL) divergence term was added to encourage statistical independence of the +different components of the latent representation, which is expected to improve its interpretability [39]. We hypothesise that it +will also contribute to producing features with higher heritability, i.e. suitable candidate phenotypes to perform GWAS on. +To train a model with such a loss function, the framework of variational autoencoder (VAE) is utilised. In this framework, +during the training phase the encoder maps the input into a probability distribution instead of a fixed vector. To emphasise this, +we will replace the notation Eθ(S) for the encoder network with qθ(Z|S), where Z is now a random variable. During training, +10/36 + +for the j-th latent variable (with 1 ≤ z j ≤ nz) two quantities are learned, µj and σ j, and a realisation zj of the random variable +Zj ∼ N (µj,σ2 +j ) is produced and passed through the decoder to generate the output mesh. The aforementioned KL-divergence +term is then used to encourage the variational approximate posterior to be a multivariate Gaussian with a diagonal covariance +structure. The regularisation term is computed as: +Ω(Strain|θ,φ) = Es∼ ˆptrain DKL +� +qθ(Z|S)||N (Z;0,1nz) +� += Es∼ ˆptrain +−1 +2nz +nz +∑ +j=1 +� +logσ2 +j −σ2 +j − µ2 +j +1 +� +(7) +where 1n is the n×n identity matrix, DKL(p||q) is the KL divergence between probability distributions p and q, and ˆptrain is +the empirical probability distribution associated with Strain. DKL(p||q) := +� p(x)ln p(x) +q(x)dp(x). The last equality in Equation 7 +arises from the formula for the KL divergence between two normal distributions where the second one is also standardised. +During testing, the mode of the latent distribution, µµµ(S), is the latent representation of the shape s. In the following, we will +rename the weighting coefficient β from Equation 4 as wKL to make it more memorable. +GWAS +According to the traditional GWAS scheme, we tested each genetic variant, Xi ∈ {0,1,2}, for association with each latent +variable zk through a univariate linear additive model of genetic effects: +zk = βikXi +εik +(8) +where εik is the component not explained by the genotype, assumed to be normally distributed. The null hypothesis tested is +that βik = 0. +Only unrelated individuals with self-reported British ancestry were included in the study, to avoid issues related to population +stratification. This produced a sample size of 31,602 individuals. Summary statistics of demographic data from these subsample +can be found in Supplementary Table S1. For the results presented in the main text, no individuals were filtered out based on +previous diagnoses or image-derived cardiac function parameters (such as ejection fraction). +Before GWAS, the phenotypes (i.e. latent variables) were adjusted for a set of covariates: sex, age, height, weight, +body-mass index, body surface area, systolic and diastolic blood pressure, alcohol consumption, smoking status and the 10 top +genomic principal components (computed within the British population). Details on how to compute the genomic principal +component loadings, as well as on the pre-processing of demographic data, are provided in the Supplementary Material. To +make this adjustment, multivariate linear regression on these covariates was performed, and then the residues of this regression +were rank-inverse-normalised. These inverse-normalised residues are the phenotypic scores to be tested in the GWAS. +It is worth mentioning that the GWAS is performed on all the individuals, including those on which the dimensionality +reduction algorithm was trained. This is correct because the algorithm does not optimise for association with genetic variants, +and therefore a uniform distribution of p-values under the null distribution can be safely assumed even when including these +subjects in the sample. +Unsupervised phenotype ensemble (UPE) +Given that the evaluation metric which guides the training, i.e. reconstruction error with a variational loss, is not necessarily +aligned with the final purpose of discovering genes that influence LV shape, there is no reason to adopt the single run with the +best value for such metric. This is the approached followed in our previous work, [6]. Indeed, the observation that a number of +loci are detected in only a small subset of runs indicates that following such a procedure would lead to failure to discover a +number of relevant genetic loci. For this reason, here we propose to adopt an ensemble-based approach, in which we pool +the different phenotypes together in a redundant yet more expressive representation. Based on the observation that different +network metaparameters, dataset partitioning and weight initialisations yielded latent representations with different genome- +wide-significant loci, we proposed to build an ensemble of phenotypes by concatenating the latent vectors for each individual +run. This composite representation provides a redundant yet more expressive representation of LV shape at end-diastole. These +runs covered a wide range of wKL, and variations in network architectures; most importantly, in the latent dimension, nz. Also, +for a given combination of metaparameters (including architecture), an optimal learning rate was found and then 5 different +random seeds were utilised to initialize the network’s weights and to partition the full dataset into train, validation and test sets +(each seed constitutes a different run). Details on the architectural parameters are given in the Supplementary Table S2. +From the full set of runs, we selected 36 training runs that achieved a good reconstruction performance: a root mean squared +deviation (RMSD) of less than 1 mm (averaged over the subjects from the test set). The deviation is taken to be the vertex-wise +11/36 + +Euclidean distance, and the mean is taken over the 5220 vertices of the LV mesh. In other words, the RMSD for subject i in run +r is: +RMSDi,r = +5220 +∑ +j=1 +� +||xi, j − ˆx(r) +i, j ||2 +2, +(9) +where xi, j denotes the triad of spatial coordinates for vertex j in the mesh of subject i, and ˆx(r) +i, j is the same for the reconstructed +mesh in run r of the autoencoder. ||·||2 +2 denotes the squared Euclidean norm. Importantly, the runs were selected based only on +mesh reconstruction error and not in the presence or absence of GWAS hits. This allows to assume a uniform distribution of +p-values over the [0,1] interval. +These 36 runs produced a total of 384 phenotypes (where the latent dimension was 8 for some runs and 16 for others). +In order to control for the false discovery rate, this procedure requires correcting the usual genome-wide Bonferroni p-value +threshold, pGW = 5 × 10−8, since the number of statistical tests that are performed grows with the size of the (pooled) +representation. In order not to overcorrect this threshold, whenever a pair of latent variables (within the same run or not) had a +Spearman correlation coefficient greater than 0.95 in absolute value, one of them was dropped at random. This procedure resulted +in 324 phenotypes to be tested in GWAS. The new study-wide threshold utilised was, therefore, pSW = pGW +324 = 1.5×10−10. +Given that for each genomic locus, the lead variant might vary across different phenotypes by virtue of high linkage +disequilibrium with close genetic variants, we adopt the following approach for locus counting: the genome is partitioned into +1703 approximately LD-independent regions, where each is region is nearly 2 megabases (Mb) in length. We compute the +number of autoencoder runs in which each region ℓ was genome-wide significant, denoting this quantity Cℓ: for each run r and +region ℓ, we retrieve the minimum p-value, across the different latent variables z(r) +k +(remember that 1 ≤ k ≤ 8 or 1 ≤ k ≤ 16, +depending on the run) which we call pℓ,r. We then count the number of runs for which pℓ,r < pGW: Cℓ = ∑R +k=1 1pℓ,r +<-CEP85L + +BRD7P3> +LOC100287632- +PLN→ +118.2 +118.4 +118.6 +118.8 +119 +Position onchr6 (Mb)Supplementary Figure S9. Triad of plots for locus TTN. +23/36 + +rs2042995,candidategeneT +(chromosome 2, position 179558366) +14 +12 +10 +10 +11 +12 +13 +14 +Chromosome +15 +100 +rs2042995 +Recombination rate (cM/Mb) +0.8 +80 +0.6 +Hog1o(p-value) +10 +0.4 +60 +0.2 +40 +0 +20 +QT interval +Blood pressure measurement (cold pressor test) +8 GWAS hits +QT interval +Late-onsetAlzheimer's +omitted +Breast size +QT interval +OSBPL6-> +DFNB59> +TTN-AS1-> +LOC101927055> +SESTD1 +HH +H +MIR548N-> +←CCDC141 +LOC101927027- + +179.2 +179.4 +179.6 +179.8 +180 +Position on chr2 (Mb)Supplementary Figure S10. Triad of plots for locus TBX5. +24/36 + +rs4767239,candidategeneTBX5 +(chromosome12,position114816548) +12 +10 +10 +11 +12 +Chromosome +100 +12 +rs4767239 +0.8 +0.6 +10 +80 +0.4 +Recombination rate (cM/Mb) +0.2 +Hog1o(p-value) +8 +60 +6 +40 +20 +QRSduration +Serum prostate-specific antigen levels +17 GWAS hits +Percent mammographic density +Night sleep phenotypes +omitted +Laryngeal squamous cell carcinoma +Colorectal cancer +RBM19 +TBX5 +←TBX3 +TBX5-AS1- +114.4 +114.6 +114.8 +115 +115.2 +Position on chr12 (Mb)Supplementary Figure S11. Triad of plots for locus GOSR2. +25/36 + +rs17608766,candidategeneGOsR2(chromosome17,position45013271） +14 +12 +10 +8 +2 +6 +8 +10 +11 +12 +1.4 +15 +16 +20 +22 +Chromosome +100 +rs17608766 +15 +0.8 +80 +Recombination rate (cM/Mb) +0.6 +Hog1o(p-value) +0.4 +10 +0.2 +60 +40 +20 +Parkinson's disease +Blood pressure +LDL cholesterol +17GWAS hits +Parkinson's disease +Cholesterol, total +omitted +Systolic blood pressure +IgG glycosylation +←ARL17A +←CDC27 +ITGB3→> +LRRC37A2→> +WNT3 +GOSR2- +MYL4- +EFCAB13- +H +王 + +MIR5089- +NSFP1 +-RPRML +44.6 +44.8 +45 +45.2 +45.4 +Position on chr17 (Mb)Supplementary Figure S12. Triad of plots for locus NKX2.5. +26/36 + +rs35564079(chromosome5，position172670611) +20 +15 +10 +8 +6 +10 +11 +12 +16 +Chromosome +12 +100 +rs35564079 +10 +O +80 +Recombination rate (cM/Mb) +Hog1o(p-value) +8 +60 +9 +40 +4 +20 +2 +0 +Adiponectin levels +14 GWAS hits +Infantile hypertrophic pyloric stenosis +Prostate cander +omitted +Periodontitis (CDC/AAP) +Height +LOC101928093-> +RPL26L1-> +CREBRF→> +NKX2-5 +MIR8056-> +LOC285593-> +←DUSP1 +LOC100268168 +BNIP1- +STC2 +←BOD1 +HIH +ERGIC1-> +LINC01484 +ATP6V0E1- +SNORA74B- +172.2 +172.4 +172.6 +172.8 +173 +Position on chr5 (Mb)Supplementary Figure S13. Triad of plots for locus LMO7. +27/36 + +10 +10 +11 +12 +Chromosome +10 +100 +13:76528303_tc_t +8 +80 +Recombinationrate (cM/Mb) +Hog10(p-value) +60 +4 +40 +20 +Type 1 diabetes +Corneal astigmatism +Diabetic retinopathy +<-TBC1D4 +LMO7-AS1 +LMO7DN> +-COMMD6 +LMO7- +UCHL3-→ +LMO7DN-/T1→ +76.2 +76.4 +76.6 +76.8 +77 +Position onchr13(Mb)Supplementary Figure S14. Triad of plots for locus RBM20. +28/36 + +rs189569984,candidategeneRBM20 +(chromosome10,position112544125) +10 +10 +11 +12 +Chromosome +10 +100 +rs189569984 +0.8 +Recombination rate (cM/Mb) +8 +80 +0.6 +0.4 +60 +0.2 +40 +20 +Crohn's disease +Fasting glucost +3 GWAS hits +Platelet aggregation +omitted +Insomnia (caffeine-induced) +MXI1-> +DUSP5- +RBM20-> +MIR4680- +ADRA2A- +←SMNDC1 +SMC3- +PDCD4-AS1 +PDCD4→ + +MTMR7 +SLC7A2→ +MTUS1 +MICU3- +←CNOT7 +PDGFRL- +VPS37A→> +-MIR548V +16.8 +17 +17.2 +17.4 +17.6 +Position on chr8 (Mb)Supplementary Figure S16. Triad of plots for locus WAC. +30/36 + +rs2797593,candidategeneWAC(chromosome10,position28655648) +12 +10 +10 +11 +12 +Chromosome +10 +100 +0.8 +rs2797593 +8 +0.6 +80 +Recombinationrate (cM/Mb) +0.4 +0.2 +Hog1o(p-value) +60 +4 +40 +20 +Pulmonary function decline +Preeclampsia +10 GWAS hits +Bone mineral density (paediatric, skull) +Percentage gas trapping +omitted +Bone mineral density +Obesity-relatedtraits +<←ARMC4 + +←LINC00837 +SNORD130 +-MIR8086 +-WAC-AS1 +LINC01517 +WAC-> +C10orf126→ +28.2 +28.4 +28.6 +28.8 +29 +Positionon chr10 (Mb)Supplementary Figure S17. Triad of plots for locus near gene LGALS8. +31/36 + +rs2853621(chromosome1,position236691532)） +12 +10 +10 +11 +12 +Chromosome +100 +rs2853621 +10 +0.8 +0.6 +80 +Recombinationrate (cM/Mb) +8 +0.4 +0.2 +Hog1o(p-value) +60 +40 +20 +chotic treatment in schizophrenia (wprking memory) +Homocysteine levels +2 GWAS hits +Periodontitis(CDC/AAP) +omitted +Urate levels (BMI interaction) + +EDARADD→ +HEATR1 +ACTN2> +MTR- +MT1HL1 +HEHHH +ERO1B +LGALS8-> +LGALS8-AS1 +236.2 +236.4 +236.6 +236.8 +237 +Position on chr1 (Mb)Supplementary Figure S18. Triad of plots for locus near gene CCDC91. +32/36 + +rs3741760(chromosome12，position28544464) +12 +10 +10 +17 +12 +Chromosome +100 +rs3741760 +12 +0.8 +0.6 +0.4 +80 +10 +0.2 +Recombinationrate +8 +60 +40 +(cM/Mb) +20 +Attention deficit hyperactivity disorder +Metabdlite levels (Pyroglutamine) +12 GWAS hits +Hyperactive-impulsive symptoms +Height +Obesity-related traits +Essential tremor +omitted +Ossification of the posterior longitudinal ligament of the spine +-PTHLH +CCDC91→ +28.2 +28.4 +28.6 +28.8 +29 +Position on chr12 (Mb)Supplementary Figure S19. Triad of plots for locus near gene EN1. +33/36 + +rs162748 (chromosome 2,position 119479427) +10 +10 +11 +12 +Chromosome +100 +rs162748 +10 +0.8 +0.6 +0.4 +80 +8 +0.2 +Recombinationrate +Hog10(p-value) +60 +9 +40 +(cM/Mb) +20 +Multiple sclerosis +Cerebrospinal T-tau levels +3 GWAS hits +ne mineral density +Verylong-chain saturatedfatty aci +omitted +Bone mineral density (spine) +←LOC101927709 +←EN1 +MARCO-> +—C1QL2 +119 +119.2 +119.4 +119.6 +119.8 +Position on chr2 (Mb)Supplementary Figure S20. Triad of plots for locus near gene BAG3. +34/36 + +rs375034445,candidategeneBAG3 +(chromosome10.position121424815） +8 +6 +10 +11 +12 +13 +14 +22 +Chromosome +10 +100 +rs375034445 +8 +C +80 +Recombination rate (cM/Mb) +Hog1o(p-value) +6 +60 +4 +C +40 +2 +20 +Dilated cardiomyopathy +IgG glycosylation +6 GWAS hits +Parkinson's disease +omitted +Type 2 diabetes +Menarche (age at onset) +<-SFXN4 +MIR4681> +RGS10 +BAG3→> +INPP5F> +SEC23IP-→ +<←PRDX3 + +CTNND1→ +OR9Q1-→> +OR10W1 +-OR5B3 +OR5B21 +HHH +MIR130A-→ +OR6Q1-> +OR9Q2- +-OR5B17 + +CLP1 +-OR1S2 +OR5B12 +ZDHHC5→> +OR10Q1 +ZFP91CNTF- +←MED19 +TMX2- +TMX2-CTNND1-→ +SELENOH +57.4 +57.6 +57.8 +58 +58.2 +Position on chr11 (Mb)Supplementary Figure S22. Triad of plots for locus STRN. +36/36 + +rs2245109,candidategeneSTRN +(chromosome2,position37086197) +10 +10 +11 +12 +Chromosome +100 +rs2245109 +10 +0.8 +80 +Recombination rate (cM/Mb) +0.6 +8 +log10(p-value) +0.4 +0.2 +60 +6 +40 +20 +gElevels inasthmatics +Chronic lymphocytic leukemi +8 GWAS hits +Schizopl +omitted +Glucose homeostasis traits +CRIM1- +VIT-> + 2: +Definition 1.4. Let (F1, . . . , Fd−1) be a tuple of tiles in Zd, d ≥ 2. We say that (F1, . . . , Fd−1) +has property (⋆) if it is an independent tuple and for every (v1, . . . , vd−1), (w1, . . . , wd−1) ∈ +F ∗ +1 × . . . × F ∗ +d−1 such that +span(v1, . . . , vd−1) = span(w1, . . . , wd−1), +we have vi = wi for all 1 ≤ i ≤ d − 2. +Theorem 1.5. Let (F1, . . . , Fd−1) be a tuple of tiles in Zd that has property (⋆). Then any +joint co-tile for F1, . . . , Fd−1 is piecewise (d − 1)-periodic. +The next statement follows immediately from Theorem 1.5 together with Theorem 1.3. +Corollary 1.6. Let (F1, . . . , Fd−1) be a tuple of tiles in Zd that has property (⋆). +If +(F1, . . . , Fd−1) admits a joint co-tile then it admits a d-periodic joint co-tile. +Note that for d = 2, property (⋆) is vacuous, hence Theorem 1.5 reduces to the statement +that any co-tile for a finite subset of Z2 is piecewise 1-periodic (Greenfeld-Tao’s theorem) +and Corollary 1.6 reduces to the statement that any finite subset of Z2 that admits a co-tile +also admits a periodic co-tile (Bhattacharya’s theorem). Hence for d ≥ 3, it is natural to ask +whether property (⋆) is a necessary condition for the existence of a periodic joint co-tile of +(d − 1) tiles of Zd . +We note a particular application of our methods, although not directly related to our main +results: +Theorem 1.7. Suppose that Zd decomposes into (d − 1)-periodic subsets A1, . . . , Ar ⊂ Zd, +where at least one of them is not d-periodic. Then there exists Γ ≤ Zd of rank d − 1 so that +Aj + Γ = Aj for all 1 ≤ j ≤ r. +On the other hand, we obtain the following converse results for Theorem 1.2 and Corol- +lary 1.6. +Theorem 1.8. Suppose that {0} ⫋ F ⋐ Zd admits a periodic tiling A ⊆ Zd, then there exist +F1, . . . , Fd−1 ⋐ Zd with 0 ∈ Fj and Fj ⊕ A = Zd for all 1 ≤ j ≤ d, such that +(a) (F1, . . . , Fd−1, F) is a d-tuple of independent tiles. +(b) (F1, . . . , Fd−2, F) has property (⋆). + +4 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +Combining Corollary 1.6 and Theorem 1.8 (b) we obtain the following: +Corollary 1.9. A finite set {0} ⫋ F ⋐ Zd tiles Zd periodically if and only if there exists +F1, . . . , Fd−2 ⋐ Zd and A ⊂ Zd such that (F1, . . . , Fd−2, F) has property (⋆), F ⊕ A = Zd and +Fj ⊕ A = Zd for all 1 ≤ j ≤ d − 2. +Remark 1.10. Note that F and A play a symmetric role in the equation F ⊕ A = Zd, A is +a co-tile for F, but F is also a co-tile for A. Assuming that F ⋐ Zd and that F ⊕ A = Zd, +the periodic tiling conjecture asks about a specific property of the set of co-tiles of F. In +view of Corollary 1.9, that property is equivalent to a property of the set of co-tiles of A. In +particular for d = 3, let F ⋐ Z3, A ⊂ Z3 such that F ⊕ A = Z3. Then F tiles Z3 periodically +if and only if there is another co-tile F ′ for A such that (F ′, F) has property (⋆). +The structure of the paper is as follows. Section 2 contains basic background and definitions. +In Section 3 we prove Theorem 3.1, a periodic decomposition theorem for joint co-tiles, which +is a refinement of Theorem 1.1. From Theorem 3.1, we directly deduce Theorem 1.1 and +Theorem 1.2. In Section 4, we discuss generalizations of Theorem 3.1, Theorem 1.1 and +Theorem 1.2 to countable abelian groups. This allows us to extend Newman’s Theorem to +tilings of the group Z × (Z/pZ). In Section 5 we prove Theorem 1.5, which asserts that +property (⋆) implies piecewise (d − 1)-periodicity of joint co-tiles. Then in Section 6 we prove +Theorem 1.7 and deduce Theorem 1.3. Section 7 is dedicated to the proof of Theorem 1.8. +Finally, Section 8 contains concluding remarks and related questions. +Acknowledgement. We thank Itay Londner for discussions about tilings in cyclic groups +and the Coven-Meyerowitz conditions. We thank Ilya Tyomkin for telling us about the +relation between the dimension of the common complex zeros for a system of multivariate +polynomials with integer coefficients, the tropical variety, and the associated Bieri-Groves set. +We also thank Rachel Greenfeld and Terrence Tao for their helpful communications. +2. Preliminaries +A function f : Zd → R is called L-periodic, where L ≤ Zd, if for every x ∈ Zd and v ∈ L +we have f(x + v) = f(x). Recall that a set A ⊆ Zd is piecewise k-periodic if A is the disjoint +union of k-periodic sets. +Definition 2.1. Let Γ1, Γ2 be abelian groups. For f : Γ1 → Γ2 and v ∈ Γ1, we define the +discrete derivative of f in direction v, Dvf : Γ1 → Γ2, by +Dvf(w) := f(w) − f(w − v). +A function P : Γ1 → Γ2 is called a polynomial map of degree at most r if +∀ v1, . . . , vr+1 ∈ Γ1 : +Dv1 . . . Dvr+1P = 0 +(where for consistency P ≡ 0 is a polynomial of degree −1). Given a subgroup Γ3 < Γ1, we +say that P : Γ1 → Γ2 is a polynomial map of degree at most r with respect to Γ3 if +∀ v1, . . . , vr+1 ∈ Γ3 : +Dv1 . . . Dvr+1P = 0. +The following basic facts about polynomials will be useful for us. Lemma 2.2 below is due +to Leibman [Lei02, Prop. 1.21]. We include a short proof for the reader’s convenience. +Lemma 2.2. Let P : Zd → R be a polynomial map with respect to a finite index subgroup +L ≤ Zd, which is bounded, then P is constant on cosets of L. + +PERIODICITY OF JOINT CO-TILES IN Zd +5 +Proof. Let r ∈ N denote the degree of P, as a polynomial with respect to L. It is clear +from Definition 2.1 that if r is equal to 0, then the restriction of P to each coset of L is a +constant. Similarly, if the degree of P is equal to 1, then the restriction of P to each coset +of L is a constant plus a non-trivial homomorphism (see e.g. [Lei02]). For contradiction, +we may assume that r ≥ 1. Observe that since P is bounded, for every v ∈ L we have +DvP ⊆ P(Zd) − P(Zd), thus DvP is bounded. Therefore, for every v1, . . . , vr−1 ∈ L the +function Dv1 . . . Dvr−1P is a bounded polynomial map of degree exactly one, with respect to +L. But non-trivial homomorphisms into R are unbounded, a contradiction. +□ +Definition 2.3. We say that a bounded function f : Zd → R has mean m if +lim +n→∞ +1 +|Bn| +� +v∈Bn +f(v) = m, +(3) +where Bn = {−n, . . . , n}d. +We say that f : Zd → R/Z is equidistributed in R/Z if +lim +n→∞ +1 +|Bn| +� +v∈Bn +g(f(v)) = +� 1 +0 +g(x)dx +(4) +holds for every continuous function g : R/Z → R, where we identify g : R/Z → R with +g : R → R such that g(x + 1) = x for all x ∈ R. +We will use the following version of Weyl’s equidistribution theorem for multivariate +polynomials, see for instance [Yif22]. +Theorem 2.4 (Weyl’s equidistribution theorem for polynomials in several variables). Let +P : Zd → R/Z be a polynomial map with respect to a finite index subgroup Γ of Zd. Then +on every coset v + Γ of Γ, the restriction of P to v + Γ is either equidistributed in R/Z or +periodic. +We implicitly rely on the following basic observation: +Proposition 2.5. Let F ⋐ Zd. Suppose that F ⊂ Bn0 for some n0 ∈ N and that f : Zd → R +is a bounded function satisfying 1F ∗ f = 1. Denote by C = |F|(max f − min f). Then for +every n > n0 one has +|Bn−n0| − C |Bn+n0 \ Bn−n0| ≤ |F| +� +w∈Bn +f(w) ≤ |Bn−n0| + C |Bn+n0 \ Bn−n0| , +(5) +and thus the function f has mean +1 +|F|. In particular, if F1, F2 ⋐ Zd satisfy 1F1 ∗f = 1F2 ∗f = 1, +then |F1| = |F2|. +Proof. Pick n0 ∈ N such that F ⊂ Bn0. Observe that 1F ∗f = 1 implies that for every n > n0 +we have +1Bn−n0 − C · 1Bn+n0\Bn−n0 ≤ 1F ∗ f|Bn ≤ 1Bn−n0 + C · 1Bn+n0\Bn−n0, +where f|Bn denotes the restriction of f to Bn. +Taking the sum of the values of these +functions over all z ∈ Zd implies that (5) holds for every n > n0. Since limn→∞ +|Bn−n0| +|Bn| += 1 +and limn→∞ +|Bn+n0\Bn−n0| +|Bn| += 0, dividing (5) by |F| · |Bn| and letting n → ∞ yields the +assertion. +□ + +6 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +Remark 2.6. The mean of a function f : Γ → R is defined similarly, using (3), for any +countable amenable group Γ, in which case Bn is replaced by a Følner sequence in Γ, and +an analogue of Proposition 2.5 holds in this more general context as well. In Section 8, +we implicitly apply Proposition 2.5 for countable abelian groups Γ, which are in particular +amenable. +2.1. Shifts of finite type. The space of co-tiles for a given finite set F ⊂ Zd, or more +generally, the space of joint co-tiles for a given collection of sets, can naturally be endued +with the structure of a compact topological space on which Zd acts by homeomorphisms. +Topological dynamical systems of this kind are called Zd-subshifts, more specifically subshifts +of finite type. We include here relevant terminology and basic facts from the field of symbolic +dynamics, particularly regarding shifts of finite type. We refer to [LM95] for a comprehensive +introduction to symbolic dynamics. +Let Σ be a finite set (alphabet) and Γ a finitely generated abelian group. The set of +functions from Γ to Σ, denoted ΣΓ, is called the full Γ-shift. For x ∈ ΣΓ and v ∈ Γ, we use xv +to denote the value of x at v (this is an element of Σ). Also for x ∈ ΣΓ and v ∈ Γ we denote +by σv(x) ∈ ΣΓ the shift of x by v, which is given by +σv(x)w = xv+w. +Endowing ΣΓ with the product topology, where the topology on Σ is the discrete topology, +makes ΣΓ a compact Γ-space. A closed, non-empty and Γ-invariant subset X ⊆ ΣΓ is called +a Γ-subshift. For x ∈ ΣΓ, the stabilizer of x is defined to be +stab(x) = {v ∈ Γ : σv(x) = x}, +which is a (possibly trivial) subgroup of Γ. A point x ∈ ΣΓ is called k-periodic if stab(x) is +a subgroup of rank k. When Γ = Z, we say that x ∈ ΣZ is periodic if it has a non-trivial +stabilizer. +Definition 2.7. A Γ-subshift X ⊆ ΣΓ is called a subshift of finite type (SFT) if there exists +a finite set W ⊂ Γ and a set F ⊆ ΣW such that +X = +� +x ∈ ΣΓ : ∀v ∈ Γ, σv(x)|W ̸∈ F +� +. +For every F ⋐ Zd the space of co-tiles for F is a subshift of finite type, under the natural +identification of the space of co-tiles for F with +XF := +� +x ∈ {0, 1}Zd : 1F ∗ x = 1 +� +. +To see that XF is indeed an SFT, take W = −F and +F = +� +p ∈ {0, 1}W : +� +w∈W +p(w) ̸= 1 +� +, +and then +XF = +� +x ∈ {0, 1}Zd : ∀v ∈ Zd, σv(x)|W ̸∈ F +� +. +Since a non-empty intersection of SFTs is also an SFT, it follows that the space of joint +co-tiles for a collection of tiles is an SFT (unless it is empty). +The following simple result is based on a pigeonhole argument. The proof is well-known +and standard, we include it for completeness. +Lemma 2.8. Every Z-subshift of finite type admits a periodic point. + +PERIODICITY OF JOINT CO-TILES IN Zd +7 +Proof. Let X ⊆ ΣZ be a Z-subshift of finite type, where Σ is a finite set. Then by definition, +there exists a finite set W ⋐ Z and F ⊆ ΣW such that +X = +� +x ∈ ΣZ : ∀v ∈ Z, σv(x)|W ̸∈ F +� +, +and X ̸= ∅. Fix x ∈ X, and let N ∈ N be an integer bigger than max(W) − min(W). Since +the set Σ{1,...,N} is finite, by the pigeonhole principle there exist integers 0 ≤ i < j ≤ |Σ|N +such that +x|{i,...,i+N−1} = x|{j,...,j+N−1}. +Let p = j − i and define ˆx ∈ ΣZ by +ˆxn = xi+(n +mod p). +Then ˆx is a periodic point, and for every n ∈ Z there exists t ∈ {i, . . . , j − 1} such that +ˆx|W+n = x|W+t. Hence, ˆx ∈ X, which proves that X admits a periodic point. +□ +We recall the following result in multidimensional symbolic dynamics. +Lemma 2.9. Let Γ be a finitely generated abelian group, Γ0 ≤ Γ a subgroup, and X ⊆ ΣΓ a +Γ-subshift. Let +XΓ0 := {x ∈ X : Γ0 ≤ stab(x)} . +(6) +If XΓ0 ̸= ∅ then it is a Γ-subshift. Furthermore, if X is a subshift of finite type then XΓ0 is +also a subshift of finite type. +Proof. First, we show that XΓ0 is a subshift. Since Γ is abelian, for every v ∈ Γ, v0 ∈ Γ0 and +y ∈ XΓ0 we have +σv0(σv(y)) = σv(σv0(y)) = σv(y). +This shows σv(y) ∈ XΓ0 for all v ∈ Γ hence XΓ0 is Γ-invariant. To see that XΓ0 is a closed +subset of ΣΓ, consider a sequence (yn)n∈N ∈ XΓ0 such that +lim +n→∞ yn = y ∈ ΣΓ +in the product topology. Since each yn ∈ XΓ0 ⊆ X and X is a closed subset of ΣΓ, we get +y ∈ X. Note that for any v0 ∈ Γ0, +σv0(y) = σv0 +� +lim +n→∞yn +� += lim +n→∞(σv0(yn)) = lim +n→∞(yn) = y, +which shows y ∈ XΓ0 and hence XΓ0 is a subshift. Now assuming that X is an SFT we show +that XΓ0 is also an SFT. Observe that XΓ0 = X ∩ Y where +Y = {x ∈ ΣΓ : Γ0 ≤ stab(x)}. +Since Γ0 is a subgroup of a finitely generated abelian group it is also finitely generated. Let +{γ1, . . . , γr} be a finite generating set for Γ0. Then +Y = +r� +i=1 +{x ∈ ΣΓ : ∀v ∈ Γ, xv+γi = xv}. +To see that Y is an SFT, let W = {0, γ1, . . . , γr} and +F = +� +w ∈ ΣW : ∃1 ≤ i ≤ r s.t. w0 ̸= wγi +� +. +Then +Y = +� +x ∈ ΣΓ : ∀v ∈ Γ, σv(x)|W /∈ F +� +. +Hence Y is an SFT, which completes the argument. + +8 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +□ +From Lemma 2.9 we deduce the following: +Lemma 2.10. Let Γ be a finitely generated abelian group of rank d. If X ⊆ ΣΓ is a Γ-subshift +of finite type that admits a (d − 1)-periodic point then it admits a d-periodic point. +Proof. Suppose X ⊆ ΣΓ is a Γ-subshift of finite type that admits a (d − 1)-periodic point, +namely a point z ∈ X and a subgroup Γ0 ≤ Γ of rank d − 1 such that stab(z) = Γ0. Let +XΓ0 be given by (6). Then XΓ0 is non-empty, and by Lemma 2.9 it is a subshift of finite +type. Because rank(Γ0) = d − 1, it follows that rank(Γ/Γ0) = 1. Let v ∈ Zd be a vector +such that k · v ̸∈ Γ0 for all k ∈ N. Then Γ0 ⊕ Zv is a finite index subgroup of Γ. Let D ⊆ Γ +be a fundamental domain for Γ0 ⊕ Zv, namely a finite set such that Γ0 ⊕ Zv ⊕ D = Γ. +Because D ⊕ Zv is a fundamental domain for Γ0 in Γ, it follows that the restriction map +ρ : XΓ0 → ΣD⊕Zv is injective, where ρ is given by ρ(x) = x |D⊕Zv. +Indeed, the inverse ρ−1 : ρ(XΓ0) → XΓ0 is given by ρ−1(˜x)u = (˜x)u′ for u ∈ Γ, where u′ is +is the unique element in (D ⊕ Zv) that satisfies u − u′ ∈ Γ0. Using the natural identification +ΣD⊕Zv ∼= (ΣD)Z, we can view ρ(XΓ0) as a subset of (ΣD)Z, which we denote by ˜X. +Let us show that ˜X is a Z-subshift of finite type. Because X is a Γ-subshift of finite +type, there exists a finite set W ⊂ Γ and F ⊂ ΣW such that XΓ0 is equal to the set of +x ∈ ΣΓ satisfying σv(x) = x and σv(x) |W̸∈ F for all v ∈ Γ0. We can assume without loss +of generality that W is a subset of Zv ⊕ D, because Zv ⊕ D is a fundmental domain for Γ0. +Let ˜W = {n ∈ Z : (nv + D) ∩ W ̸= ∅}. Then W = � +n ˜ +W(W ∩ (nv + D)). Thus, there is a +natural bijection between ΣW and (ΣD) ˜ +W. Let ˜F denote the image of F under this bijection. +Then it follows directly that +˜X = +� +x ∈ (ΣD)Z : ∀v ∈ Z : σv(x) | ˜ +W̸∈ ˜F +� +. +This proves that ˜X is indeed a Z-subshift of finite type. +Since ˜X is a Z-subshift of finite type, by Lemma 2.8 there exists a periodic point ˜z in ˜X. +Let x = ρ−1(˜z), then x ∈ X is a d-periodic point. +□ +3. The periodic decomposition theorem +The following theorem asserts a certain decomposition for a joint co-tile of k-tuple of tiles in +Zd. The case where k = 1 and f is {0, 1}-valued essentially coincides with [GT21a, Theorem +1.7], which is closely related to [Bha20, Theorem 3.3]. In the particular case that the tuple of +tiles is independent, Theorem 1.1 is a direct consequence. Namely, the indicator function of +any joint co-tile of k independent tiles is a sum of k-periodic functions, each taking values +in [0, 1]. The goal of this section is to prove the periodic decomposition theorem for joint +co-tiles and to deduce Theorem 1.1 and Theorem 1.2. +Theorem 3.1 (Periodic decomposition theorem). Let F1, . . . , Fk ⋐ Zd, with 0 ∈ Fi for all +1 ≤ i ≤ k, and let f : Zd → Z be a bounded function that satisfies 1Fi ∗f = 1 for all 1 ≤ i ≤ k. +We denote by S := |F1| = . . . = |Fk| (see Proposition 2.5). Then for every 1 ≤ i ≤ k and +every (v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i there exists a function φv1,...,vi : Zd → [min f, max f] with +the following properties: + +PERIODICITY OF JOINT CO-TILES IN Zd +9 +(a) For i < k we have +φv1,...,vi = 1 − +� +vi+1∈F ∗ +i+1 +φv1,...,vi,vi+1. +(b) +f = (−1)i +� +(v1,...,vi)∈F ∗ +1 ×...×F ∗ +i +φv1,...,vi + +i +� +j=1 +(−(S − 1))j−1. +(c) Let q denote the product of all primes less than or equal to (max f − min f)S, then +(Zqv1 + . . . + Zqvi) ≤ stab(φv1,...,vi), +(d) 1Fj ∗ φv1,...,vi = 1 for all 1 ≤ j ≤ k. In particular, φv1,...,vi has mean 1/S. +There are various extensions of Theorem 3.1. Some of these generalizations have further +applications. For the sake of readability, we do not state the most general form and instead +indicate certain generalizations in the following sections, at the expense of some repetition. +The proof of Theorem 3.1 relies on Lemma 3.2 below. Various versions of this lemma, +which is referred to as the dilation lemma, have been proved in [GT21a, Lemma 3.1], [Bha20, +Proposition 3.1] for Γ = Zd, d ≥ 1. We also refer our readers to [Tij95, Theorem 1] where this +lemma is proved for integers. The proof is based on some elementary commutative algebra +and it easily extends to countable abelian groups. For the sake of self-containment, we include +a sketch of the proof below. The proof below is nearly identical to [GT21a, Lemma 3.1], +except that we apply the assumption that r is co-prime to the order of torsion elements +directly before eq. (7). +Lemma 3.2 (Dilation lemma). Let Γ be a countable abelian group. Let 0 ∈ F ⋐ Γ, ℓ ∈ N +and f : Γ → Z a bounded function satisfying +1F ∗ f = ℓ. +Let q1 be the product of all primes less than or equal to (max f − min f)|F|, let q2 be the +product of all the orders of the torsion elements in (F − F), and set q = q1q2. Then +1rF ∗ f = ℓ, +for all r ∈ N such that r = 1 mod q. +Proof. We use the notation f ∗p = f ∗ . . . ∗ f +� +�� +� +×p +. For any prime p we have +1∗p +F = +�� +v∈F +δv +�∗p += +� +v∈F +δ∗p +v +mod p, +where the last equality holds by the Frobenius identity (f + g)∗p = f ∗p + g∗p mod p. For +integers p that are co-prime to q2 we have that p(v1 − v2) ̸= 0 for any v1 ̸= v2 ∈ F, so the +function v �→ pv is injective on F. Thus: +� +v∈F +δ∗p +v = +� +v∈F +δpv = 1pF. +(7) + +10 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +Now convolving both sides of 1F ∗ f = ℓ by 1∗(p−1) +F +yields 1∗p +F ∗ f = ℓ|F|p−1. Combining +the above, for primes p that are co-prime to q2 we obtain 1pF ∗ f = ℓ|F|p−1 mod p. If +additionally p is co-prime to |F| by Fermat little theorem |F|p−1 = 1 mod p, thus +1pF ∗ f = ℓ +mod p. +Note that both 1F ∗f and 1pF ∗f take values in [|F| min f, |F| max f]. Recall that ℓ = 1F ∗f, +so ℓ ∈ [|F| min f, |F| max f]. Thus, for p that is also greater than the size of that interval, +the above equality holds without the +mod p, namely 1pF ∗ f = ℓ. Finally, for r = 1 mod q, +r is a product of primes that satisfy the conditions above, and the result follows by iterating +the equation 1pF ∗ f = ℓ. +□ +Proof of Theorem 3.1. For 1 ≤ i ≤ k, (v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i and N ∈ N denote: +φ(N) +v1,...,vi := 1 +N i +N +� +n1,...,ni=1 +δ(1+n1q)v1+...+(1+niq)vi ∗ f. +(8) +Let q be the product of all primes less than or equal to (max f − min f)S. By applying +Lemma 3.2 for Fj with Γ = Zd and ℓ = 1 we get 1rFj ∗ f = 1 for every r ∈ qN + 1. Since +0 ∈ Fj we obtain +f = 1 − +� +v∈F ∗ +j +δrv ∗ f for every 1 ≤ j ≤ k. +For every N ∈ N, setting r = 1 + nq for n ∈ {1, . . . , N} and taking average we conclude that +for every 1 ≤ j ≤ k we have +f = 1 − +� +v∈F ∗ +j +1 +N +N +� +n=1 +δ(1+nq)v ∗ f. +(9) +Since φ(N) +v1 += 1 +N +�N +n=1 δ(1+nq)v1 ∗ f this gives (with j = 1): +f = 1 − +� +v1∈F ∗ +1 +φ(N) +v1 . +(10) +For 1 ≤ i < k, choose any (v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i and 1 ≤ n1, . . . , ni ≤ N. Setting +j = i + 1 in (9) and convolving both sides of the equation by δ(1+n1q)v1+...+(1+niq)vi we obtain +δ(1+n1q)v1+...+(1+niq)vi ∗ f = 1 − +� +vi+1∈F ∗ +i+1 +1 +N +N +� +ni+1=1 +δ(1+n1q)v1+...+(1+niq)vi+(1+ni+1q)vi+1 ∗ f. +By averaging over 1 ≤ n1, . . . , ni ≤ N and applying the definition in (8) we obtain that +φ(N) +v1,...,vi = 1 − +� +vi+1∈F ∗ +i+1 +1 +N i+1 +N +� +n1,...,ni+1=1 +δ(1+n1q)v1+...+(1+ni+1q)vi+1 ∗ f = 1 − +� +vi+1∈F ∗ +i+1 +φ(N) +v1,...,vi+1. +(11) +Since |F ∗ +i | = S − 1 for 1 ≤ i ≤ k, using (10), (11) and an inductive argument we obtain +that for every N ∈ N and 1 ≤ i ≤ k we have +f = +i +� +j=1 +(−(S − 1))j−1 + (−1)i +� +(v1,...,vi)∈F ∗ +1 ×...×F ∗ +i +φ(N) +v1,...,vi +(12) + +PERIODICITY OF JOINT CO-TILES IN Zd +11 +Notice that the functions δ(1+n1q)v1+...+(1+niq)vi ∗ f are bounded between min f and max f, +thus by (8), the functions φ(N) +v1,...,vi are bounded between min f and max f for every (v1, . . . , vi) ∈ +F ∗ +1 × . . . × F ∗ +i . In particular, for every (v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i the sequence of functions +(φN +v1,...,vi)N∈N is uniformly bounded, hence by Arzel`a–Ascoli theorem (or by a Cantor diagonal- +ization argument), it converges along a subsequence. We denote the limit by φv1,...,vi. Then +for every (v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i we have +min f ≤ φv1,...,vi ≤ max f, +and in view of (11) and (12) we have achieved (a) and (b). +To see (c), using (8), a standard telescoping argument shows that for every w ∈ Zd, +v = (v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i and every 1 ≤ j ≤ i we have +��φ(N) +v1,...,vi(w + qvj) − φ(N) +v1,...,vi(w) +�� ≤ 2N k−1 +N k += 2 +N . +Thus for every (v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i the function φv1,...,vi is qvj-periodic for every +1 ≤ j ≤ i. +It is left to see (d). +Clearly, since 1Fj ∗ f = 1, for every 1 ≤ i, j ≤ k, +(v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i and n1, . . . , ni ∈ N we have 1Fj ∗ (δ(1+n1q)v1+...(1+niq)vi ∗ f) = 1. +Thus, by (8), 1Fj ∗ φ(N) +v1,...,vi = 1 for every N ∈ N and therefore 1Fj ∗ φv1,...,vi = 1 for every +1 ≤ i, j ≤ k. In particular, by Proposition 2.5, φv1,...,vi has mean 1/S. +□ +Remark 3.3. Under the assumption that f is {0, 1}-valued, it directly follows from The- +orem 3.1, part (a), that for every 1 ≤ i < k and every (v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i , the +sum � +vi+1∈F ∗ +i+1 φv1,...,vi,vi+1 is a [0, 1]-valued function. Theorem 3.1, where k = 1 and f is +{0, 1}-valued, coincides with [GT21a, Theorem 1.7]. We will not make use of the property +that 1Fj ∗ φv1,...,vi = 1 in this paper. We mention it only for completeness and possibly for +future reference. The fact that the functions φv1 each have mean 1/S played an implicit role +in [Bha20]. +Using the assumption that the tuple of tiles is independent Theorem 1.1 is an immediate +corollary of Theorem 3.1, with f being a {0, 1}-valued function. The proof of Theorem 1.2 is +straightforward. +Proof of Theorem 1.2. Suppose that (F1, . . . , Fd) is an independent tuple of tiles in Zd and +that f : Zd → Z is a bounded function satisfying 1Fi ∗ f = 1 for all 1 ≤ i ≤ d. By +Proposition 2.5, we have |F1| = . . . = |Fd| := S. Let q be the product of all primes less than +or equal to (max f − min f)S and let +L = +� +(v1,...,vd)∈F ∗ +1 ×...×F ∗ +d +qZv1 + . . . + qZvd. +Apply Theorem 3.1 with k = d. It follows that f is a sum of functions whose stabilizers are +rank d-subgroups, more precisely, +f = (−1)d +� +(v1,...,vd)∈F ∗ +1 ×...×F ∗ +d +φv1,...,vd + +d +� +j=1 +(−(S − 1))j−1, +and for each (v1, . . . , vd) ∈ F ∗ +1 × . . . × F ∗ +d we have that qZv1 + . . . + qZvd ≤ stab(φv1,...,vd). By +the above, stab(f) contains the intersection of stab(φv1,...,vd) over (v1, . . . , vd) ∈ F ∗ +1 × . . . × F ∗ +d , +that in turn contains L. + +12 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +By the assumption that the tuple (F1, . . . , Fd) is independent, qZv1 + . . . + qZvd is a finite +index subgroup of Zd for every (v1, . . . , vd) ∈ F ∗ +1 × . . . × F ∗ +d . Since L is an intersection of +finitely many finite index subgroups, L is also a finite index subgroup. This proves that f is +periodic. +□ +4. Joint co-tilings in finitely generated abelian groups +It is natural to ask which of the results about tilings generalize from Zd to more general +groups. An inspection of the proof of Theorem 3.1 reveals that the statement still holds, +and the same proof applies, if we replace Zd by an arbitrary countable abelian group Γ, and +change the value of q in Theorem 3.1 (c) by multiplying it with the product of the orders of +all torsion elements in F − F. +There is a simple observation that allows one to reduce statements about tilings of countable +abelian groups by a finite set to the finitely generated case: Let Γ be a countable abelian +group and let F ⋐ Γ with 0 ∈ F. Let Γ0 denote the group generated by the difference set +F − F. The assumption 0 ∈ F implies that F ⋐ Γ0. Then for any co-tile A of F we have that +A ∩ Γ0 is a co-tile of F in Γ0, and tilings of Γ by F decompose into tilings of cosets of Γ0 in +Γ. A corresponding statement is true also for a tuple of tiles (F1, . . . , Fk) and a joint co-tile. +Recall that g1, . . . , gk in a countable abelian group Γ are called independent if the equation +�k +j=1 njgj = 0, with n1, . . . , nk ∈ Z, implies that n1 = . . . = nk = 0. With this definition, +Theorem 1.1 extends directly as follows: +Theorem 4.1. Let Γ be a countable abelian group. For every k ∈ N the indicator function of +any joint co-tile for k independent tiles in Γ is equal, up to a constant, to a sum of [0, 1]-valued +functions whose stabilizer has rank at least k. +Similarly, Theorem 1.2 extends as follows: +Theorem 4.2. Let Γ be a finitely generated abelian group of rank d. Any joint co-tile for d +independent tiles in Γ has a finite orbit. +A quick remark about the condition of independence for a tuple of tiles for finitely generated +abelian groups with non-trivial torsion: If Γ is of the form Γ = Zd × G where G is a finite +abelian group and (F1, . . . , Fk) is an independent tuple of tiles in Γ, then the only torsion +element in each of the sets Fi is 0. For this reason, Newman’s theorem (i.e. any tiling of Z +by a finite set is periodic) does not hold in abelian groups Γ that are finite extensions of Z. +Indeed, take Γ = Z × G, where G is a finite abelian group. Take F = {1} × G ⋐ Γ, then the +co-tiles of F are all the sets A ⊂ Γ of the following form: +A = {(n, gn) : n ∈ Z}, +for some sequence (gn)n∈Z of elements in G. In particular, it is no longer true that any co-tile +of F must be periodic, unless G is trivial. Nonetheless, if G is a finite cyclic group of prime +order, then the only obstructions to extending Newman’s theorem are of this form. +Proposition 4.3. If Γ = Z × (Z/pZ) for some prime number p and F ⋐ Γ is a finite set, +then every co-tile of F is periodic, unless F is of the form F = ˜F × (Z/pZ) for some finite +tile ˜F ⋐ Z, in which case the co-tiles of F are all of the form +A = {(n, gn) : n ∈ ˜A}, gn ∈ Z/pZ, +(13) +where ˜A is a co-tile of ˜F ⋐ Z, which by Newman’s theorem must be periodic. + +PERIODICITY OF JOINT CO-TILES IN Zd +13 +The proof of the proposition relies on the following generalization of Theorem 3.1. +Theorem 4.4. Let Γ be a countable abelian group, F1, . . . , Fk ⋐ Γ such that |Fi| = S, +and 0 ∈ Fi for all 1 ≤ i ≤ k, and let f : Γ → Z be a bounded function that satisfies +1Fi ∗ f = 1 for all 1 ≤ i ≤ k. For every 1 ≤ i ≤ k, let F Tor +i +denote the intersection of +Fi with the torsion subgroup of Γ, and let F ∗ +i = Fi \ F Tor +i +. Then for every 1 ≤ i ≤ k and +every (v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i there exists a function φv1,...,vi : Γ → [min f, max f] with +the following properties: +(a) For i < k we have +1F Tor +i+1 ∗ φv1,...,vi = 1 − +� +vi+1∈F ∗ +i+1 +φv1,...,vi,vi+1. +(b) For every 1 ≤ i ≤ k there is an integer constant Ci such that +1F Tor +1 +∗ . . . ∗ 1F Tor +i +∗ f = (−1)i +� +(v1,...,vi)∈F ∗ +1 ×...×F ∗ +i +φv1,...,vi + Ci. +(c) Let q1 be the product of all primes less than or equal to (max f − min f)S, let q2 be +the product of all the orders of the torsion elements in the sets Fi − Fi, for 1 ≤ i ≤ k, +and set q = q1q2. Then +(Zqv1 + . . . + Zqvi) ≤ stab(φv1,...,vi), +(d) 1Fj ∗ φv1,...,vi = 1 for all 1 ≤ j ≤ k. In particular, φv1,...,vi has mean 1/S. +The proof of Theorem 4.4 below is a minor adaptation of the proof of Theorem 3.1. Note +that in the case where Γ is a torsion free abelian group, F Tor +i += {0}. In particular, when +Γ = Zd, Theorem 4.4 coincides with Theorem 3.1. +Proof. By applying Lemma 3.2 for Fi with ℓ = 1 and q as in (c) we get 1rFi ∗ f = 1 for every +r ∈ qN + 1. Because r = 1 mod q, we have rF Tor +i += F Tor +i +. Since Fi = F Tor +i +⊎ F ∗ +i we have +1F Tor +i +∗ f = 1 − +� +v∈F ∗ +i +δrv ∗ f for every 1 ≤ i ≤ k. +For every N ∈ N, setting r = 1 + nq for n ∈ {1, . . . , N} and taking average we conclude that +for every 1 ≤ j ≤ k we have +1F Tor +j +∗ f = 1 − +� +vj∈F ∗ +j +1 +N +N +� +nj=1 +δ(1+njq)vj ∗ f. +(14) +Applying (14) with j = i + 1, convolving both sides by δ(1+n1q)v1+...+(1+niq)vi and taking +average over +1 +Ni +�N +n1,...,ni=1 yields +1F Tor +i+1 ∗ +� +1 +N i +N +� +n1,...,ni=1 +δ(1+n1q)v1+...+(1+niq)vi ∗ f +� += +1 − +� +vi+1∈F ∗ +i+1 +1 +N i+1 +N +� +n1,...,ni,ni+1=1 +δ(1+n1q)v1+...+(1+niq)vi+(1+ni+1q)vi+1 ∗ f. + +14 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +Defining φ(N) +v1,...,vi = +1 +Ni +�N +n1,...,ni=1 δ(1+n1q)v1+...+(1+niq)vi ∗ f, as in (8), we obtain +1F Tor +i+1 ∗ φ(N) +v1,...,vi = 1 − +� +vi+1∈F ∗ +i+1 +φ(N) +v1,...,vi,vi+1. +(15) +Note that (14) with j = 1 becomes 1F Tor +1 +∗ f = 1 − � +v1∈F ∗ +1 φ(N) +v1 . Convolving both sides by +1F Tor +2 +and using (15) with i = 1 gives +1F Tor +1 +∗ 1F Tor +2 +∗ f = |F Tor +2 +| − +� +v1∈F ∗ +1 +1F Tor +2 +∗ φ(N) +v1 += |F Tor +2 +| − +� +v1∈F ∗ +1 +� +�1 − +� +v2∈F ∗ +2 +φ(N) +v1,v2 +� +� . +By an inductive argument we obtain that for every N ∈ N and 1 ≤ i ≤ k there is a constant +Ci ∈ Z, that does not depend on N, such that +1F Tor +1 +∗ . . . 1F Tor +i +∗ f = Ci + (−1)i +� +(v1,...,vi)∈F ∗ +1 ×...×F ∗ +i +φ(N) +v1,...,vi. +(16) +Items (a) and (b) follow from (15) and (16) respectively. The rest of the proof is completely +identical to the proof of Theorem 3.1 and therefore omitted. +□ +Lemma 4.5. Let p be a prime number and let ∅ ̸= F0 ⫋ Z/pZ. Then 1F0 is an invertible +element of the ring QZ/pZ, where multiplication in the ring is convolution. In other words, +there exists g ∈ QZ/pZ such that g ∗ 1F0 = δ0. +Proof. Consider the ring Q[x]/⟨xp − 1⟩ (with operations of addition and multiplication of +polynomials). It is easy to check that this ring is isomorphic as a ring to QZ/pZ, with the +operations of pointwise addition and convolution. The isomorphism is given by identifying +an element +p−1 +� +i=0 +aixi + ⟨xp − 1⟩ ∈ Q[x]/⟨xp − 1⟩ +with the function f ∈ QZ/pZ given by f(i + pZ) = ai. +Let F0 ⊂ Z/pZ be a non-empty proper subset of Z/pZ. Then 1F0 ∈ QZ/pZ is naturally +identified with the coset of the polynomial P(x) = � +(i+pZ)∈F0 xi in Q[x]/⟨xp − 1⟩. Then the +assumption that F0 is a non-empty proper subset of Z/pZ implies that the polynomial P +is co-prime to the cyclotomic polynomial of order p, Φp = �p−1 +i=0 xi. Since P(1) = |F0| ̸= 0 +it follows that P is co-prime to x − 1. Because xp − 1 = Φp(x)(x − 1), it follows that P is +co-prime to xp − 1. Hence there exists polynomials Q1, Q2 ∈ Q[x] such that +1 = Q1(x)P(x) + Q2(x)(xp − 1). +This means that in the ring Q[x]/⟨xp − 1⟩, the coset of Q1(x)P(x) is the same as the coset +of the polynomial 1. Since the coset of the polynomial 1 in Q[x]/⟨xp − 1⟩ corresponds to +δ0 ∈ QZ/pZ, this implies that g ∗ 1F0 = δ0, where g ∈ QZ/pZ is the element corresponding to +the coset of Q1. +□ +Proof of Proposition 4.3. Let p be a prime number and F ⋐ Z × (Z/pZ) be a finite set. +Suppose A ⊂ Z × (Z/pZ) satisfies 1F ∗ 1A = 1. Applying Theorem 4.4 with Γ = Z × (Z/pZ) +k = 1, F1 = F and f = 1A, we conclude that 1F Tor ∗ 1A is a sum functions having infinite +stabilizer, hence 1F Tor ∗ 1A is periodic. + +PERIODICITY OF JOINT CO-TILES IN Zd +15 +First, assume that there is a set ˜F ⋐ Z such that F = ˜F×Z/pZ. So 1F = 1 ˜F×{0}∗1{0}×(Z/pZ). +Thus 1 ˜F×{0} ∗ 1{0}×(Z/pZ) ∗ 1A = 1. This implies that 1{0}×(Z×pZ) ∗ 1A ≤ 1, so for every n ∈ Z +there exists at most one element gn ∈ Z/pZ such that (n, gn) ∈ A. Hence, in this case, A is +of the form (13) for some set ˜A ⊂ Z. It follows that 1 ˜F ∗ 1 ˜ +A = 1, where the convolution here +is with respect to the group Z. +Now suppose that F is not of the above form. This means that there exists n ∈ Z such +that F ∩ ({n} × Z/pZ) is a non-empty proper subset of {n} × (Z/pZ). By translating F we +can assume without loss of generality that F Tor is neither empty nor equal to {0} × (Z/pZ). +Then there exists a non-empty proper subset F0 ⊂ Z/pZ such that F Tor = {0} × F0. In this +case, by Lemma 4.5, there exists g : Z/pZ → Q such that g ∗ 1F0 = δ0, where the convolution +is in (Z/pZ). Let ˜g : Z × Z/pZ → Q be given by ˜g(0, i) = g(i) for i ∈ Z/pZ and g(n, i) = 0 +for every n ∈ Z \ {0} and i ∈ Z/pZ. Then ˜g ∗ 1F Tor = δ0, where this time the convolution is +in Z × (Z/pZ). Since 1F Tor ∗ 1A is periodic, so is ˜g ∗ 1F Tor ∗ 1A = 1A. +We have thus shown that in the case that F is not of the form F = ˜F × (Z/pZ) for some +˜F ⋐ Z, every co-tile is periodic. +□ +5. Property (⋆) implies (d − 1)-piecewise periodicity +In this section, we use property (⋆) to deduce Theorem 1.5. To this end, we will use +Theorem 2.4, which is a version of Weyl’s equidistribution theorem for polynomials in several +variables. +The relevance of Weyl’s equidistribution theorem to our setting comes from +Lemma 5.1 below. We note that similar arguments have appeared earlier in [Bha20], [KS20] +and [GT21a]. +Lemma 5.1. Suppose g, g1, . . . , gm : Γ1 → Γ2 are functions, where Γ1, Γ2 are abelian groups, +such that �m +i=1 gi = g. Suppose g is a polynomial of degree at most r ∈ N with respect to a +subgroup Γ0 ≤ Γ1. For any 1 ≤ i < j ≤ m define the group Li,j = stab(gi) + stab(gj), and let +L = � +1≤i 1, take v ∈ L, then in particular v ∈ L1,2 ∩ Γ0 and thus v = v1 + v2 for some +v1 ∈ stab(g1) and v2 ∈ stab(g2). Note that for every function f : Γ1 → Γ2, the identity +Dvf = Dv1f ◦ σv2 + Dv2f holds, where σu : Γ1 → Γ1 denotes the shift by u, σu(w) = w − u. +Since Dv1g1 = 0, applying this identity to g1 = − �m +i=2 gi + g yields +Dvg1 = Dv2g1 = −Dv2 +� m +� +i=2 +gi − g +� +. +Since Dv2g2 = 0 we have +Dvg1 + +m +� +i=3 +Dv2gi = Dv2g. +(17) +Note that v2 ∈ Γ0, hence Dv2g is a polynomial of degree at most r − 1 with respect to Γ0. So +by the induction hypothesis, each summand on the left-hand side in (17) is a polynomial of +degree at most max{m − 2, r − 1} with respect to a subgroup L′, defined in a similar way + +16 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +to L using the functions Dvg1, Dv2g3, . . . , Dv2gm. In particular, for every v ∈ L the function +Dvg1 is a polynomial of degree at most max{m − 2, r − 1} with respect to L′. +Now observe that for every f : Γ1 → Γ2 and v ∈ Γ1 we have stab(f) ⊆ stab(Dvf), thus +L ≤ L′ and for every v ∈ L we, in particular, have that Dvg1 is a polynomial of degree at +most max{m − 2, r − 1} with respect to L. In a similar way for 2 ≤ i ≤ m and every v ∈ L, +each Dvgi is a polynomial of degree at most max{m − 2, r − 1} with respect to L, which +completes the proof. +□ +Lemma 5.2. Suppose g : Zd → [0, 1] is a function such that: +(1) g mod 1 is a polynomial with respect to a finite index subgroup of Zd. +(2) g is a sum of finitely many non-negative (d − 1)-periodic functions. +Then there exists a finite index subgroup Γ ≤ Zd such that the restriction of g to each coset +of Γ is (d − 1)-periodic. +Proof. Suppose g = �m +i=1 gi, where gi : Zd → [0, 1] and rank(stab(gi)) ≥ d − 1. In case that +rank (�m +i=1 stab(gi)) ≥ d − 1, the function g is (d − 1)-periodic and the assertion follows. +Otherwise, by summing together some of the gi’s we can assume without loss of generality +that stab(gi) + stab(gj) is a finite index subgroup of Zd, for every i ̸= j. By Lemma 5.1, +because g modulo 1 is a polynomial with respect to a finite index subgroup, we conclude that +each of the gi’s modulo 1 are polynomials with respect to a finite index subgroup Γ0 ≤ Zd. +Let +Γ = Γ0 ∩ +� +i̸=j +(stab(gi) + stab(gj)) . +We will show that g is (d − 1)-periodic on each coset of Γ. Since Γ ≤ Γ0, each gi modulo 1 is +also a polynomials with respect to Γ. Hence by Weyl’s equidistribution theorem (Theorem 2.4), +every gi modulo 1 is either equidistributed or periodic, on each coset of Γ. +Fix u ∈ Zd. Let g(u) : (u + Γ) → [0, 1] denote the restriction of g to this coset. We consider +3 cases: +(1) Suppose there exists 1 ≤ i ≤ m and v ∈ (u + Γ) such that gi(v) = 1. Then because +0 ≤ g(v) ≤ 1 and gj(v) ≥ 0, we conclude that gj(v) = 0 for all j ̸= i. But gi(v) = 1 +implies that gi(v+w1) = 1 for all w1 ∈ stab(gi) so by the same argument gj(v+w1) = 0 +for all w1 ∈ stab(gi). Thus, gj(v + w1 + w2) = 0 for all w1 stab(gi) and w2 ∈ stab(gj). +Since Γ ≤ stab(gi) + stab(gj), we conclude that gj is zero on the coset u + Γ, for +all j ̸= i. This shows that in this case g(u) = gi on u + Γ, and in particular g(u) is +(d − 1)-periodic. So in the remaining cases we can assume that none of the gi’s are +equal to one, hence the gi’s obtain values in the interval [0, 1). +(2) Suppose there exists 1 ≤ i ≤ m such that gi is equidistributed modulo 1 on u + Γ. Let +0 < ϵ < 1 be smaller than all the non-zero values obtained by the (possibly empty) +set of gj that are periodic modulo 1. Because gi is equidistributed modulo 1 on u + Γ, +there exists v ∈ u + Γ such that gi(v) > 1 − ϵ. Thus, gj(v) < ϵ for all j ̸= i. As in +the previous part, using Γ ≤ stab(gi) + stab(gj), we conclude that gj(w) < ϵ for all +j ̸= i and all w ∈ u + Γ. This tells us that in particular that gj is not equidistributed +modulo 1 on u + Γ. By the choice of ϵ, gj(w) = 0 for every periodic j ̸= i and every +w ∈ u + Γ. We conclude also in this case that g = gi on u + Γ and particular g(u) is +(d − 1)-periodic. + +PERIODICITY OF JOINT CO-TILES IN Zd +17 +(3) The remaining case is that all the gi’s modulo 1 are periodic on u + Γ, but since they +take values in [0, 1), the gi’s themselves are all d-periodic. It follows in this case that +gu is d-periodic, as the sum of d-periodic functions (and in particular (d− 1)-periodic). +□ +Proof of Theorem 1.5. We conveniently assume d > 2, because the case d = 2 is covered by +[GT21a]. Suppose that A ⊂ Zd satisfies Fi⊕A = Zd for all 1 ≤ i ≤ d−1, where (F1, . . . , Fd−1) +is a tuple of tiles in Zd that has property (⋆), see Definition 1.4. Let φv1,...,vd−1 : Zd → [0, 1] be +as in Theorem 3.1, applied for k = d − 1 and f = 1A. Given (v1, . . . , vd−2) ∈ F ∗ +1 × . . . × F ∗ +d−2 +and a (d − 1)-dimensional subspace V < Rd such that v1, . . . , vd−2 ∈ V , define +ψV = +� +wd−1∈F ∗ +d−1∩V +φv1,...,vd−2,wd−1. +Note that by the independence of (F1, . . . , Fd−1), every (d − 1)-tuple in F ∗ +1 × . . . × F ∗ +d−1 +spans a (d − 1)-dimensional subspace. Denote by H the set (counted without multiplicity) of +all (d − 1)-dimensional subspaces of Rd spanned by (d − 1)-tuples in F ∗ +1 × . . . × F ∗ +d−1, and +for (v1, . . . , vd−2) ∈ F ∗ +1 × . . . × F ∗ +d−2 let H(v1, . . . , vd−2) ⊂ H be the set of such subspaces +of dimension (d − 1) that contain v1, . . . , vd−2. Thus, for every fixed tuple (v1, . . . , vd−2) ∈ +F ∗ +1 × . . . × F ∗ +d−2 we have +� +wd−1∈F ∗ +d−1 +φv1,...,vd−2,wd−1 = +� +V ∈H(v1,...,vd−2) +ψV . +(18) +By property (⋆), {H(v1, . . . , vd−2) : (v1, . . . , vd−2) ∈ F ∗ +1 × . . . × F ∗ +d−2} is a partition of H, +therefore +� +(v1,...,vd−1)∈F ∗ +1 ×...×F ∗ +d−1 +φv1,...,vd−1 = +� +V ∈H +ψV . +(19) +It follows that the functions ψV possess the following three properties: +(i) +1 − φv1,...,vd−2 = +� +V ∈H(v1,...,vd−2) +ψV . +(ii) stab(ψV ) is a rank (d − 1) subgroup of V ∩ Zd. +(iii) ψV modulo 1 is a polynomial with respect to a finite index subgroup of Zd. +Indeed, property (i) is a direct consequence of Theorem 3.1 part (a) with i = d − 1, combined +with (18). Property (ii) follows from Theorem 3.1 part (c). Setting � +ψV = ψV +mod 1, the +equation in Theorem 3.1 part (b) (with f = 1A and i = d − 1), combined with (19), yields +that � +V ∈H � +ψV = 0. By property (ii), stab(� +ψV ) + stab(� +ψV ′) is a finite index subgroup of Zd +whenever V, V ′ ∈ H and V ̸= V ′. Thus property (iii) follows from Lemma 5.1. +In view of these three properties, Lemma 5.2 can be applied to g = 1 − φv1,...,vd−2, for any +(v1, . . . , vd−2) ∈ F ∗ +1 × . . . × F ∗ +d−2. This implies that there is a finite index subgroup Γd−2 ≤ Zd +such that each φv1,...,vd−2 is a polynomial with respect to Γd−2, and its restriction to every +coset u + Γd−2 is (d − 1)-periodic. +Next, we iterate the above argument using the recursion formula in part (a) of Theorem 3.1 +combined with Lemma 5.2. In turn, this yields a finite index subgroup Γ1 ≤ Zd such that + +18 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +each φv1 is a polynomial with respect to Γ1, and its restriction to every coset u + Γ1 is +(d − 1)-periodic. By part (b) of Theorem 3.1 with i = 1 we have that +1 − 1A = +� +v1∈F ∗ +1 +φv1. +So applying Lemma 5.2 to g = 1 − 1A, we obtain a finite index subgroup Γ ≤ Zd such that +the restriction of 1 − 1A to each coset of Γ is (d − 1)-periodic. Hence the restriction of 1A to +each coset of Γ is (d − 1)-periodic. Thus, if u1, . . . , ur are cosets representatives of Γ in Zd, +setting Aui = A ∩ (ui + Γ) ⊂ Zd yields a decomposition A = Au1 ⊎ . . . ⊎ Aur of A into finitely +many (d − 1)-periodic sets, as required. +□ +6. From piecewise (d − 1)-periodicity to d-periodicity +The following lemma extracts an idea that appears within the proof of [GT21a, Theorem +5.4]. +Lemma 6.1. Suppose that f1, . . . , fr, f : Zd → R are bounded functions satisfying f = +�r +i=1 fj. Assume additionally that: +(1) stab(fi) + stab(fj) is a finite index subgroup of Zd for all 1 ≤ i < j ≤ r. +(2) stab(f) is a finite index subgroup of Zd. +Then, for each 1 ≤ j ≤ r, the group stab(fj) is of finite index in Zd. +Proof. Let g1 = f1−f and gj = fj for 2 ≤ j ≤ r. Then g1+. . .+gr = 0 and stab(gi)+stab(gj) +is a finite index subgroup of Zd for all 1 ≤ i < j ≤ r. Using the fact that 0 is a polynomial, +and applying Lemma 5.1, we get that each gi is a polynomial with respect to a finite index +subgroup of Zd. But each gi is bounded. By Lemma 2.2, a polynomial with respect to a +finite index subgroup of Zd that is bounded must be constant on cosets of this finite index +subgroup. This implies that for each 1 ≤ j ≤ r the group stab(fj) is of finite index in Zd. +□ +Theorem 1.7 is a direct consequence of the above lemma, as shown below. +Proof of Theorem 1.7. Set fj = 1Aj, then �r +j=1 fj = 1. Let Lj ≤ Zd be the subgroups of rank +at least d − 1 that stabilizes Aj. Note that for every two such subgroups Lj1, Lj2 ≤ Zd, either +their intersection has rank d − 1 or their sum has finite index in Zd. Assume by contradiction +that the intersection of all Lj’s is of rank less than d − 1. By unifying some of the Aj’s we +can assume without loss of generality that Lj1 + Lj2 is a finite index subgroup of Zd for all +1 ≤ i < j ≤ r. In this case, the conditions of Lemma 6.1 hold but the conclusion fails, by the +initial assumption. Thus the assumption that rank +��r +j=1 Lj +� +< d − 1 is false. +□ +We would also need the following lemma. +Lemma 6.2. Suppose that Σ ⋐ R is a finite set of real numbers, g1, . . . , gr : Zd → R are +finitely supported functions and f : Zd → Σ is a (d − 1)-periodic function such that gj ∗ f is +d-periodic for every 1 ≤ j ≤ r. Then there exists a d-periodic function ˜f : Zd → Σ such that +gj ∗ f = gj ∗ ˜f for every 1 ≤ j ≤ r. +Proof. Consider the space +X = {˜x ∈ ΣZd : ∀1 ≤ j ≤ r, gj ∗ ˜x = gj ∗ f}, + +PERIODICITY OF JOINT CO-TILES IN Zd +19 +and let Γ = �r +j=1 stab(gj ∗ f). Then X is a Γ-shift of finite type, and by definition f ∈ X is +a (d − 1)-periodic point in X. Apply Lemma 2.10 to conclude that there exists ˜f ∈ X that is +d-periodic. Any such point ˜f satisfies the conclusion of the lemma. +□ +At this stage, we are prepared to present the proof of Theorem 1.3. +Proof of Theorem 1.3. Suppose that A ⊆ Zd is a piecewise (d − 1)-periodic joint co-tile for +F1, . . . , Fk ⋐ Zd. That is, there exists functions f1, . . . , fr : Zd → {0, 1}, each fj is (d − 1)- +periodic, and 1A = �r +j=1 fj. Notice that we may assume that rank +�� +j stab(fj) +� +< d − 1. +Indeed, if rank +�� +j stab(fj) +� +≥ d − 1 then 1A = �r +j=1 fj is a (d − 1)-periodic point in the +shift of finite type �k +i=1 Tile(Fi; Zd), and thus by Lemma 2.10 it contains a d-periodic point. +Also note that for every two subgroups L1, L2 ≤ Zd having rank at least d − 1, either their +intersection has rank at least d − 1 or L1 + L2 has finite index in Zd. So as before, by possibly +summing some of the fj’s we can assume without loss of generality that stab(fl) + stab(fj) +is a finite index subgroup of Zd for all 1 ≤ l < j ≤ r. Now consider the functions 1Fi ∗ fj. +Observe that for every 1 ≤ i ≤ k we have �r +j=1 1Fi ∗ fj = 1, and for every 1 ≤ j ≤ r we have +stab(fj) ≤ stab(1Fi ∗fj). Thus setting Λi,j := stab(1Fi ∗fj) yields that rank (Λi,j) ≥ d−1 and +Λi,l + Λi,j is a finite index subgroup of Zd, for every 1 ≤ i ≤ k and 1 ≤ l < j ≤ r. Applying +Lemma 6.1 for each 1 ≤ i ≤ k separately we see that each Λi,j is a finite index subgroup of +Zd. That is, each one of the functions 1Fi ∗ fj is d-periodic. For any fixed 1 ≤ j ≤ r, applying +Lemma 6.2 with gi = 1Fi and f = fj and Σ = {0, 1}, yields a d-periodic function ˜fj : Zd → Σ +that satisfies 1Fi ∗ ˜fj = 1Fi ∗ fj, for all 1 ≤ i ≤ k. In particular, the function f : Zd → Z +defined by f := �r +j=1 ˜fj is bounded, d-periodic, and it satisfies +∀1 ≤ i ≤ k : +1Fi ∗ f = 1Fi ∗ +� +r +� +j=1 +˜fj +� += +r +� +j=1 +1Fi ∗ ˜fj = +r +� +j=1 +1Fi ∗ fj = 1. +Since ˜f := �r +j=1 ˜fj is a sum of {0, 1}-valued functions and 1Fi ∗ f = 1, it follows that f itself +is {0, 1}-valued, hence ˜A is an indicator of a set ˜A such that Fi ⊕ ˜A = Zd. Since each ˜fj is +d-periodic, so is ˜A. This completes the proof. +□ +7. Constructing independent tiles with the 1-hyperplane repetition +property for a periodic co-tile +In this section, we prove Theorem 1.8. We repeatedly rely on the following basic fact. +Lemma 7.1. Let L ≤ Zd be a finite index subgroup and let U1, . . . , Ur ⊂ Rd be affine +subspaces of dimension strictly smaller than d. Then the set L \ �r +i=1 Ui is infinite. +Proof. For n ∈ N let Bn = {−n, . . . , n}d. +Then there exist c, c1, . . . , cr > 0 such that +|Bn ∩ L| ≥ cnd while |Bn ∩ Ui| ≤ cindim Ui ≤ cind−1. In particular, |Bn ∩ (L \ �r +i=1 Ui)| tends +to infinity as n tends to infinity. +□ +Lemma 7.2. Let F ⋐ Zd, let A ⊆ Zd such that F ⊕ A = Zd and let L ≤ Zd be a subgroup +satisfying A + L = A. Then for every function f : F → L the tile set +Ff := {v + f(v) : v ∈ F} +satisfies Ff ⊕ A = Zd. + +20 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +Proof. Given a function f : F → L, we show that Ff ⊕ A = Zd. The condition F ⊕ A = Zd +can be rewritten as Zd = � +v∈F(v + A). Since A + L = A and f(v) ∈ L for every v ∈ F, it +follows that f(v) + A = A. Thus, +Zd = +� +v∈F +(v + A) = +� +v∈F +(v + f(v) + A) = +� +˜v∈Ff +(˜v + A). +This proves that Ff ⊕ A = Zd. +□ +Lemma 7.3. Suppose we are given d, m ∈ N, (v1, . . . , vm) ∈ Zd and a finite index subgroup +L ≤ Zd. Given a subset J ⊆ {1, . . . , m} and a subspace W < Rd, let VW(g, J) denote the +subspace of Rd/W obtained by projecting span{vj + g(j) : j ∈ J} into Rd/W via the map +v �→ v + W. Then for every finite collection W of proper subspaces of Rd there exists a +function g : {1, . . . , m} → L so that for every J ⊆ {1, . . . , m} and every W ∈ W we have +dim (VW(g, J)) = min{d − dim(W), |J|}. +Proof. We prove the claim by induction on m. For m = 1, we only need to choose g(1) ∈ L such +that v1+g(1) ̸∈ W for any W ∈ W. This is possible by Lemma 7.1. Assume by induction that +g(1), . . . , g(m) ∈ L have been defined so that the conclusion holds for every J ⊆ {1, . . . , m} +and every W ∈ W. Using Lemma 7.1 we can choose g(m+1) ∈ L that is not contained in any +affine hyperplane of the form U := −vm+1 +span{vj +g(j) : j ∈ J}+W, where W ∈ W and +J ranges over subsets of {1, . . . , m} of size at most d − dim(W) − 1. We need to show that +for any J ⊆ {1, . . . , m + 1} and W ∈ W we have dim (VW(g, J)) = min{d − dim(W), |J|}. +Fix some J ⊆ {1, . . . , m + 1} and W ∈ W. +The assertion follows from the induction +hypothesis in case (m + 1) ̸∈ J, so suppose (m + 1) ∈ J. By the induction hypothesis, +dim (VW(g, J \ {m + 1})) = min {d − dim(W), |J \ {m + 1}|}. If |J \{m+1}| ≥ d−dim(W), +then dimW(V (g, J)) = d − dim(W), as required. Otherwise, we have that +dim(VW(g, J \ {m + 1})) = |J \ {m + 1}| = |J| − 1. +By our choice of g(m+1), we have that vm+1+g(m+1) ̸∈ span {vj + g(vj) : j ∈ J \ {m + 1}}, +so +dimW(V (g, J)) = dim(VW(g, J \ {m + 1})) + 1 = |J|. +This completes the induction step, hence the proof. +□ +Proof of Theorem 1.8. Suppose F ⊕A = Zd where L ∈ Zd is a finite index subgroup satisfying +A + L = A. Write F ∗ = {w1, . . . , wk}. We apply Lemma 7.3 with m = (d − 1)k and +(v1, . . . , vm), where vkj+i = wi for 0 ≤ j ≤ d−2, and 1 ≤ i ≤ k, and W = {span{v} : v ∈ F} +to obtain a function g : {1, . . . , m} → L as in the statement of Lemma 7.3. For 0 ≤ j ≤ d − 2 +we set +Fj+1 = {0} ∪ {vkj+i + g(kj + i) : 1 ≤ i ≤ k} = {0} ∪ {wi + g(kj + i) : 1 ≤ i ≤ k}. +By Lemma 7.2 we indeed have Fj ⊕ A = Zd for every 1 ≤ j ≤ d − 1. To see that +(F1, . . . , Fd−1, F) is a d-tuple of independent tiles, note that for any choice of (u1, . . . , ud−1, v) ∈ +F ∗ +1 × . . . × F ∗ +d−1 × F there exists i1, . . . , id−1 ∈ {1, . . . , k} so that +uj = vk(j−1)+ij + g(k(j − 1) + ij). +Hence, there exists a set J ⊂ {1, . . . , k(d − 1)} so that +span{u1 + W, . . . , ud−1 + W} = VW(g, J), + +PERIODICITY OF JOINT CO-TILES IN Zd +21 +where W = span{v}. By the property of g, it follows that dim(span{u1, . . . , ud−1, v}) = d. +Let us check that {F1, . . . , Fd−2, F} has the property (⋆). Choose two distinct (d−2)-tuples +(u1, . . . , ud−2), (˜u1, . . . , ˜ud−2) ∈ F ∗ +1 × . . . × F ∗ +d−2, +and v, ˜v ∈ F. As before, it follows that there exists subsets J, ˜J ⊂ {1, . . . , m} with J ̸= ˜J +and |J| = | ˜J| = d − 2 so that +{u1, . . . , ud−2} = {vj : j ∈ J} and {˜u1, . . . , ˜ud−2} = {vj : j ∈ J′}. +Since J ̸= ˜J and |J| = | ˜J| = d − 2, there exists ℓ ∈ ˜J \ J. It follows from the property of the +function g that for any v ∈ F ∗ +dim(span({vj : j ∈ J} ∪ {v}) = d − 1 +and +dim(span({vj : j ∈ J} ∪ {v} ∪ {vℓ}) = d. +This shows that vℓ ̸∈ span ({vj : j ∈ J} ∪ {v}). In particular, there does not exist v ∈ F ∗ +such that +span ({˜u1, . . . , ˜ud−2}) ⊆ span ({u1, . . . , ud−2, v}) . +This shows that there does not exist v, ˜v ∈ F ∗ so that +span ({˜u1, . . . , ˜ud−2, ˜v}) = span ({u1, . . . , ud−2, v}) , +which proves that (F1, . . . , Fd−2, F) has property (⋆). +□ +8. Further comments and questions +8.1. Integer-valued co-tiles. Given F ⋐ Γ, we say that a bounded function f : Γ → Z is +an integer-valued co-tile for F if 1F ∗ f = 1. Observe that our proof of Theorem 1.3 holds for +integer-valued co-tile as well, thus we have: +Proposition 8.1. Let k and d be positive integers and let F1, . . . , Fk ⋐ Zd. Suppose that +F1, . . . , Fk admit an integer-valued joint co-tile f and that f = �r +i=1 fr, where each fi : Zd → +Z is bounded and (d − 1)-periodic. Then F1, . . . , Fk admit a d-period integer-valued joint +co-tile. +It is natural to ask whether the existence of an integer-valued co-tile for F ⋐ Γ implies +the existence of a set A ⊆ Γ for which 1F ∗ 1A = 1? The simple example below shows that +this is not true even for Γ = Z (or for Γ a finite cyclic group, here Z/18Z). Let F1 = {0, 1}, +F2 = {0, 3, 6} and F = F1 ⊕ F2 = {0, 1, 3, 4, 6, 7}. +We claim that F does not tile Z, but it does admit an integer-valued co-tile. Note that for +A1 = 2Z and A2 = {0, 1, 2} ⊕ 9Z we have +F1 ⊕ A1 = F2 ⊕ A2 = Z. +Furthermore, if ˜A1 is a co-tile for F1 then ˜A1 must be a translate of A1. To see that F does +not tile Z, suppose by contradiction that F ⊕ A = Z then F1 ⊕ (F2 ⊕ A) = Z, so we must +have that F2 ⊕ A is a coset of 2Z, but this is clearly impossible since F2 is not contained in a +coset of 2Z. Now take +f = 1A1 − 1A2. + +22 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +Then using 1F = 1F1 ∗ 1F2 and 1Fi ∗ 1 = |Fi| we get: +1F ∗ f = 1F2 ∗ (1F1 ∗ 1A1) − 1F1 ∗ (1F2 ∗ 1A2) = |F2| − |F1| = 1. +8.2. Conditions for joint tilings for d independent tiles in Zd. In view of Theorem 1.2, +the classical Wang argument (see [Ber66], [Rob71]) implies that it is algorithmically decidable +whether a set of d independent tiles in Zd admit a joint co-tile: Indeed, any such tiling +must be periodic so we can exhaust the possible periodic co-tiles. As in [GT21a], from an +upper bound for the period of a co-tile one can directly deduce an upper bound for the +computational complexity of the tiling problem. It is of interest to find explicit necessary +and sufficient conditions for a d-tuple of independent subsets of Zd to admit a joint co-tile. +In view of Theorem 1.8, the previous problem is closely related to the more basic question of +finding explicit necessary and sufficient conditions for a finite set of Zd to tile periodically. +Conversely, one can ask about necessary and sufficient conditions for an infinite subset of +Zd to be a joint co-tile for d-independent tiles. In view of Theorem 1.2 and Theorem 1.8 +this is equivalent to the question of finding necessary and sufficient conditions for a periodic +subset of Zd to be a co-tile for a finite tile. +A complete solution to the above questions involves the factorization of finite abelian +groups, namely understanding solutions for A ⊕ B = G, where G is a finite abelian group. +This is a difficult problem even in the cyclic case G = Z/MZ, which comes up in tilings of Z. +Coven and Meyerowitz [CM99] found explicit and efficiently verifiable sufficient conditions +for tiling the integers by a finite set. It has been conjectured that these conditions are also +necessary. This conjecture has been verified in some specific cases recently [�LL22a, �LL22b]. +The necessity of the Coven-Meyerowitz conditions would imply an efficient algorithm for +determining if a given finite subset F ⋐ Z can tile Z, see [KM09]. +8.3. Higher level tilings. A level ℓ co-tile of Zd by a finite set set F ⋐ Zd is a set A ⊆ Zd +such that 1F ∗ 1A = ℓ. Both Theorem 1.1 and Theorem 1.2 generalize to level ℓ tilings. A +suitable modification of Proposition 2.5 implies that if 1F ∗ f = ℓ then f has mean +ℓ +|F|. A +proof can be obtained via a relatively routine modification of Theorem 3.1 as follows: +Theorem 8.2. Let ℓ1, . . . , ℓk ∈ N, F1, . . . , Fk ⋐ Zd, with 0 ∈ Fi for all 1 ≤ i ≤ k, and let +f : Zd → Z be a bounded function that satisfies 1Fi ∗ f = ℓi for all 1 ≤ i ≤ k. Then for +every 1 ≤ i ≤ k and every (v1, . . . , vi) ∈ F ∗ +1 × . . . × F ∗ +i there exists a function φv1,...,vi : Zd → +[min f, max f] with the following properties: +(a) For i < k we have +φv1,...,vi = ℓi+1 − +� +vi+1∈Fi+1 +φv1,...,vi,vi+1. +(b) +f = (−1)i +� +(v1,...,vi)∈F ∗ +1 ×...×F ∗ +i +φv1,...,vi + +i +� +j=1 +(−1)j−1 +j� +t=1 +ℓt +j−1 +� +s=1 +|Fs|. +(c) Let q denote the product of all primes less than or equal to max1≤i≤k ℓk(max f − +min f) max1≤i≤k |Fi|, then +(Zqv1 + . . . + Zqvi) ≤ stab(φv1,...,vi), +(d) 1Fj ∗ φv1,...,vi = ℓi for every 1 ≤ j ≤ k. In particular, it has mean ℓi/|Fi|. + +PERIODICITY OF JOINT CO-TILES IN Zd +23 +8.4. Piecewise 1-periodicity of co-tiles in Z2 × (Z/pZ). By applying the arguments of +Section 4, the methods of [GT21a] directly give: +Theorem 8.3. Let p be a prime number, Γ = Z2 × (Z/pZ) and F ⋐ Γ be a finite set. Then +one of the following holds: +(1) Any A ⊂ Γ satisfying F ⊕ A = Γ is piecewise 1-periodic. +(2) There exist a finite set ˜F ⊂ Z2 such that F = ˜F × (Z/pZ). +In fact, using Theorem 4.4 and the results of Section 5, we can deduce the following: For +any rank 2 abelian group Γ and any F ⋐ Γ, if F ⊕ A = Γ then the set F Tor ⊕ A is piecewise +1-periodic, where as in Section 4, F Tor is the intersection of F with the torsion subgroup of Γ. +Then in the case Γ = Z2 × Z/pZ with p prime, Lemma 4.5 implies Theorem 8.3. +Corollary 8.4. Let p be a prime number, Γ = Z2 × (Z/pZ) and F ⋐ Γ be a finite set. If F +tiles Γ, then F tiles Γ periodically. +Rachel Greenfeld and Terrence Tao have informed us in private communication that they +also obtained Corollary 8.4. +8.5. A Fourier-analytic and algebraic-geometric approach. Fourier analytic methods +are a natural approach to translational tiling problems, see [GT21a, Remark 1.8]. +Let +g1, . . . , gd : Zd → C be finitely supported functions, by which we mean that gi(v) = 0 for all +but finitely many v ∈ Zd. Suppose f : Zd → C is a bounded function that satisfies gi ∗ f = 1 +for all 1 ≤ i ≤ d. Taking distributional Fourier transform on both sides yields +ˆgi · ˆf = δ0. +Thus, the distributional Fourier transform of f is supported on 0 and the intersection of the +zeros of ˆgi. In particular, if ˆg1, . . . , ˆgd have finitely many common zeros, and f must be the +Fourier transform of a multivariate trigonometric polynomial, hence periodic. +The set of common zeros for d polynomials in d variables is “generically” a finite set. +Given v = (n1, . . . , nd) ∈ Zd ++ let Xv := xn1 +1 · . . . · xnd +d denote the corresponding monomial in d +variables x1, . . . , xd. Given a finite set F ⋐ Zd ++, let PF := � +v∈F Xv denote the corresponding +multivariate polynomial. We conclude that whenever F1, . . . , Fd ⋐ Zd ++ are subsets such that +the algebraic variety +V (PF1, . . . , PFd) := +d� +i=1 +� +(x1, . . . , xd) ∈ Cd : PFi(x1, . . . , xd) = 0 +� +has a finite intersection with the d-sphere, then any joint co-tile for F1, . . . , Fd is periodic. +This raises the question: Is it true that for an independent d-tuple (F1, . . . , Fd) in Zd the +algebraic variety V (PF1, . . . , PFd) is finite? +We note that it can be shown that V (PF1, . . . , PFd) is finite if we impose the somewhat +stronger condition that (F1 − F1, . . . , Fd − Fd) is an independent d-tuple in Zd. This follows +from the equality of the tropical variety with the Bieri-Groves set of the variety (see Theorem +2.2.5 and Corollary 2.2.6 in [EKL06]), combined with [EKL06, Theorem 2.2.3] and an explicit +direct computation. This connection was kindly explained to us by Ilya Tyomkin. This +argument gives an alternative derivation of the conclusion of Theorem 1.2, under the slightly +stronger assumption that (F1 − F1, . . . , Fd − Fd) is an independent tuple of tiles on Zd. + +24 +TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON +References +[Ber66] +R. Berger. The undecidability of the domino problem. Mem. Amer. Math. Soc., 66, 1966. +[Bha20] +S. Bhattacharya. Periodicity and decidability of tilings of Z2. Amer. J. Math., 142(1):255–266, 2020. +[BN91] +D. Beauquier and M. Nivat. On translating one polyomino to tile the plane. Discrete Comput. +Geom., 6(6):575–592, 1991. +[CM99] +E. Coven and A. Meyerowitz. Tiling the integers with translates of one finite set. J. Algebra, +212(1):161–174, 1999. +[EKL06] Manfred Einsiedler, Mikhail Kapranov, and Douglas Lind. Non-Archimedean amoebas and tropical +varieties. J. Reine Angew. Math., 601:139–157, 2006. +[GS87] +B. Gr¨unbaum and G. C. Shephard. Tilings and patterns. W. H. Freeman and Company, New York, +1987. +[GT21a] R. Greenfeld and T. Tao. The structure of translational tilings in Zd. Discrete Anal., 16:1–28, 2021. +[GT21b] R. Greenfeld and T. Tao. Undecidable translational tilings with only two tiles, or one nonabelian +tile. arXiv:2108.07902, 2021. +[GT22] +R. Greenfeld and T. Tao. A counterexample to the periodic tiling conjecture. arXiv:2211.15847, +2022. +[Ken92] +R. Kenyon. Rigidity of planar tilings. Invent. Math., 107(3):637–651, 1992. +[Khe22] +A. Khetan. A periodicity result for tilings of Z3 by clusters of prime-squared cardinality. +arXiv:2109.14179, 2022. +[KM09] +M. N. Kolountzakis and M. Matolcsi. Algorithms for translational tiling. J. Math. Music, 3(2):85–97, +2009. +[KS20] +J. Kari and M. Szabados. An algebraic geometric approach to Nivat’s conjecture. Inform. and +Comput., 271:104481, 25, 2020. +[Lei02] +A. Leibman. Polynomial mappings of groups. Israel J. Math., 129:29–60, 2002. +[�LL22a] +I. �Laba and I. Londner. The coven–meyerowitz tiling conditions for 3 odd prime factors. Invent. +math., 10.1007/s00222-022-01169-y, 2022. +[�LL22b] +I. �Laba and I. Londner. Splitting for integer tilings and the coven-meyerowitz tiling conditions. +arXiv:2207.11809v1, 2022. +[LM95] +D. Lind and B. Marcus. An introduction to symbolic dynamics and coding. Cambridge University +Press, Cambridge, 1995. +[LW96] +J. C. Lagarias and Y. Wang. Tiling the line with translates of one tile. Invent. Math., 124:341–365, +1996. +[New77] D. J. Newman. Tesselation of integers. J. Number Theory, 9(1):107–111, 1977. +[Rob71] +R. M. Robinson. Undecidability and nonperiodicity for tilings of the plane. Invent. Math., 12:177–209, +1971. +[Sze98] +M. Szegedy. Algorithms to tile the infinite grid with finite clusters. In Proceedings 39th Annual +Symposium on Foundations of Computer Science (Cat. No.98CB36280), pages 137–145, 1998. +[Tij95] +R. Tijdeman. Decomposition of the integers as a direct sum of two subsets. In Number theory (Paris, +1992–1993), volume 215 of London Math. Soc. Lecture Note Ser., pages 261–276. Cambridge Univ. +Press, Cambridge, 1995. +[WvL84] H. A. G. Wijshoff and J. van Leeuwen. Arbitrary versus periodic storage schemes and tessellations +of the plane using one type of polyomino. Inform. and Control, 62(1):1–25, 1984. +[Yif22] +Y. Yifrach. A note about weyl equidistribution theorem. arXiv:2201.07138, 2022. +Ben-Gurion University of the Negev, Department of Mathematics, Beer-Sheva, 8410501, +Israel. mtom@bgu.ac.il +Ben-Gurion University of the Negev, Department of Mathematics, Beer-Sheva, 8410501, +Israel. sanadhya@post.bgu.ac.il +Ben-Gurion University of the Negev. Department of Mathematics. Beer-Sheva, 8410501, +Israel. yaars@bgu.ac.il + diff --git a/KNFIT4oBgHgl3EQfaCs0/content/tmp_files/load_file.txt b/KNFIT4oBgHgl3EQfaCs0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..37e9d4d4e2f8428eb53bc8e001550102498f4962 --- /dev/null +++ b/KNFIT4oBgHgl3EQfaCs0/content/tmp_files/load_file.txt @@ -0,0 +1,1721 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf,len=1720 +page_content='PERIODICITY OF JOINT CO-TILES IN Zd TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Given finite subsets F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk in Zd, a joint co-tile is a set A ⊆ Zd that satisfies Fj ⊕ A = Zd for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We study the structure of joint co-tiles in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We introduce a notion of independence for a tuple of finite subsets of Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We prove that for any d ≥ 1, any joint co-tile for d independent sets is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' This generalizes a classical result of Newman stating that any tiling of Z by a finite set is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For a (d − 1)-tuple of finite subsets of Zd that satisfy a certain technical condition that we call property (⋆), we prove that any joint co-tile decomposes into disjoint (d − 1)-periodic sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Consequently, we show that for a (d − 1)-tuple of finite subsets of Zd that satisfy property (⋆), the existence of a joint co-tile implies the existence of periodic joint co-tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' These results are generalizations to higher dimensions of Bhattacharya’s theorem (the proof of the periodic tiling conjecture for Z2) and Greenfeld-Tao’s theorem about the structure of co-tiles in Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Conversely, we prove that if a finite subset F in Zd admits a periodic co-tile A, then there exist (d − 1) additional tiles that together with F are independent and admit A as a joint co-tile, and (d − 2) additional tiles that together with F satisfy the property (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Combined, our results give a new necessary and sufficient condition for a finite subset of Zd to tile periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We also discuss tilings and joint tilings in other countable abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Introduction For a countable abelian group Γ we write F ⋐ Γ to indicate that F is a finite subset of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For A ⊆ Γ we denote by F ⊕ A = � a∈A (F + a), where the notation of the right-hand side stands for a disjoint union of the sets {F + a}a∈A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The notation F ⊕ A = E thus means that every e ∈ E has a unique representation as e = f +a where f ∈ F and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We say that F tiles Γ if there exists a collection of disjoint union of translates of F whose union is equal to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' That is, F tiles Γ if there exists a set A ⊆ Γ such that F ⊕ A = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (1) In that case, we say that A is a co-tile for the tile F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let g, h : Γ → R, where Γ is a countable abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We denote by g ∗ h the usual convolution function given by g ∗ h(x) = � y∈Γ g(y) · h(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Using this notation, equation (1) is equivalent to 1F ∗ 1A = 1, where 1X denotes the indicator function of the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 52C22, 37B52, 05B45, 52C23, 37B10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' tranlational tilings, periodic tiling conjecture, tiling equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='11255v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='DS] 26 Jan 2023 2 TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON Let Γ be a countable abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Elements g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , gk ∈ Γ are called independent if the only integers n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , nk ∈ Z that satisfy �k j=1 njgj = 0 are n1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' = nk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Recall that the rank of an abelian group is the maximal size of an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Suppose that Γ is an abelian group of rank d and that k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' A set C ⊆ Γ is called k-periodic if there exists a subgroup L ≤ Γ, with rank(L) ≥ k, such that C + L = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In the case that k = d we will also say that C is periodic instead of d-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We say that a tile set F ⋐ Γ tiles Γ periodically if there exits a periodic co-tile for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' If F tiles Γ but does not admit a periodic co-tile, then the set F is called aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Newman [New77] proved that any tiling of Γ = Z by a finite set is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Already for Γ = Z2, it is not difficult to find tilings of Γ by a finite set that are not even 1-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' See [GT21a, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='3] for some examples and a brief discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Still, it is natural to ask for different generalizations of Newman’s theorem to higher-rank abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' It has been conjectured for some time that for any F ⋐ Zd, if there exists A ⊆ Zd such that F ⊕ A = Zd then there exists a periodic A′ ⊆ Zd such that F ⊕ A′ = Zd [LW96], [GS87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' This conjecture became known as the periodic tiling conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The periodic tiling conjecture can be interpreted as an attempt to generalize Newman’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The Z2 case of the periodic tiling conjecture was proved several years ago by Bhattacharya [Bha20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Other instances of the periodic tiling conjecture have been proved, under additional assumptions [BN91, Khe22, Ken92, Sze98, WvL84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The periodic tiling conjecture has recently been disproved for sufficiently large d by Greenfeld and Tao [GT22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In this paper, we study the structure of sets A ⊆ Zd that satisfy Fj ⊕ A = Zd for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , k, (2) for subsets F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk ⋐ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We refer to such an A as a joint co-tile for F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In [GT21b], sets A ⊆ Zd satisfying (2) have been referred to as solutions to the system of tiling equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' As with ordinary systems of linear equations, it makes sense to introduce a notion of independence in this setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For F ⋐ Zd we denote F ∗ := F \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We say that (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk) is an independent tuple of tiles (or k independent tiles) if each Fj is a finite subset of Zd, with 0 ∈ Fj, and for every choice of v1 ∈ F ∗ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vk ∈ F ∗ k , the k-tuple (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vk) is independent (equivalently here, linearly independent vectors over Q, or similarly over R or C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Notice that if (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk) is an independent tuple of tiles then k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Observe that the existence of a joint co-tile for F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk ⋐ Zd implies that |F1| = |F2| = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' = |Fk| (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Building on methods developed in [Bha20], [GT21a] and earlier work, we prove the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For every 1 ≤ k ≤ d, the indicator function of any joint co-tile for k independent tiles in Zd is equal, up to a constant, to a sum of [0, 1]-valued k-periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The case k = 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 was proven in [GT21a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' As a consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1, we obtain the following generalization of Newman’s result for any dimension: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Any joint co-tile for d independent tiles in Zd is d-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Furthermore, if 1Fi ∗ f = 1 holds for d independent tiles (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd) and a bounded function f : Zd → Z, then f is d-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We discuss further generalizations of Newman’s theorem in Section 4 and particularly to the group Z × (Z/pZ) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' PERIODICITY OF JOINT CO-TILES IN Zd 3 We say that a set A ⊆ Zd is piecewise k-periodic if there exist A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Ar ⊂ Zd such that A = �r j=1 Aj and each Aj is k-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Note that [Bha20] and [GT21a] used weakly periodic for piecewise 1-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In [GT21a] it was shown that any A ⊆ Z2 satisfying F ⊕ A = Z2 is piecewise 1-periodic, whereas in [Bha20] it was shown that almost every solution to F ⊕ A = Z2 is piecewise 1-periodic, with respect to any invariant measure on the space of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The apriori weaker “almost everywhere” result sufficed to prove the Z2 periodic tiling conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The following result shows that the existence of piecewise (d − 1)-periodic joint co-tiles implies the existence of d-periodic joint co-tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For k = 1 and d = 2 it coincides with the results in [Bha20], [GT21a], deducing 2-periodicity from piecewise 1-periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let k and d be positive integers and let F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk ⋐ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' If F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk admit a piecewise (d − 1)-periodic joint co-tile, then they admit a d-period joint co-tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We now define an additional condition on a tuple of tiles, that is needed for the formulation of a certain generalization of Bhattacharya’s and Greenfeld-Tao’s theorems to d > 2: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−1) be a tuple of tiles in Zd, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We say that (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−1) has property (⋆) if it is an independent tuple and for every (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vd−1), (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , wd−1) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ d−1 such that span(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vd−1) = span(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , wd−1), we have vi = wi for all 1 ≤ i ≤ d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−1) be a tuple of tiles in Zd that has property (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then any joint co-tile for F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−1 is piecewise (d − 1)-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The next statement follows immediately from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5 together with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−1) be a tuple of tiles in Zd that has property (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' If (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−1) admits a joint co-tile then it admits a d-periodic joint co-tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Note that for d = 2, property (⋆) is vacuous, hence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5 reduces to the statement that any co-tile for a finite subset of Z2 is piecewise 1-periodic (Greenfeld-Tao’s theorem) and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='6 reduces to the statement that any finite subset of Z2 that admits a co-tile also admits a periodic co-tile (Bhattacharya’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Hence for d ≥ 3, it is natural to ask whether property (⋆) is a necessary condition for the existence of a periodic joint co-tile of (d − 1) tiles of Zd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We note a particular application of our methods, although not directly related to our main results: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Suppose that Zd decomposes into (d − 1)-periodic subsets A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Ar ⊂ Zd, where at least one of them is not d-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then there exists Γ ≤ Zd of rank d − 1 so that Aj + Γ = Aj for all 1 ≤ j ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' On the other hand, we obtain the following converse results for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2 and Corol- lary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Suppose that {0} ⫋ F ⋐ Zd admits a periodic tiling A ⊆ Zd, then there exist F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−1 ⋐ Zd with 0 ∈ Fj and Fj ⊕ A = Zd for all 1 ≤ j ≤ d, such that (a) (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−1, F) is a d-tuple of independent tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (b) (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−2, F) has property (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 4 TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON Combining Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='6 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='8 (b) we obtain the following: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' A finite set {0} ⫋ F ⋐ Zd tiles Zd periodically if and only if there exists F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−2 ⋐ Zd and A ⊂ Zd such that (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd−2, F) has property (⋆), F ⊕ A = Zd and Fj ⊕ A = Zd for all 1 ≤ j ≤ d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Note that F and A play a symmetric role in the equation F ⊕ A = Zd, A is a co-tile for F, but F is also a co-tile for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Assuming that F ⋐ Zd and that F ⊕ A = Zd, the periodic tiling conjecture asks about a specific property of the set of co-tiles of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In view of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='9, that property is equivalent to a property of the set of co-tiles of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In particular for d = 3, let F ⋐ Z3, A ⊂ Z3 such that F ⊕ A = Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then F tiles Z3 periodically if and only if there is another co-tile F ′ for A such that (F ′, F) has property (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Section 2 contains basic background and definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In Section 3 we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1, a periodic decomposition theorem for joint co-tiles, which is a refinement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1, we directly deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In Section 4, we discuss generalizations of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2 to countable abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' This allows us to extend Newman’s Theorem to tilings of the group Z × (Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In Section 5 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5, which asserts that property (⋆) implies piecewise (d − 1)-periodicity of joint co-tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then in Section 6 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='7 and deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Section 7 is dedicated to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Finally, Section 8 contains concluding remarks and related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We thank Itay Londner for discussions about tilings in cyclic groups and the Coven-Meyerowitz conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We thank Ilya Tyomkin for telling us about the relation between the dimension of the common complex zeros for a system of multivariate polynomials with integer coefficients, the tropical variety, and the associated Bieri-Groves set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We also thank Rachel Greenfeld and Terrence Tao for their helpful communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Preliminaries A function f : Zd → R is called L-periodic, where L ≤ Zd, if for every x ∈ Zd and v ∈ L we have f(x + v) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Recall that a set A ⊆ Zd is piecewise k-periodic if A is the disjoint union of k-periodic sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let Γ1, Γ2 be abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For f : Γ1 → Γ2 and v ∈ Γ1, we define the discrete derivative of f in direction v, Dvf : Γ1 → Γ2, by Dvf(w) := f(w) − f(w − v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' A function P : Γ1 → Γ2 is called a polynomial map of degree at most r if ∀ v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vr+1 ∈ Γ1 : Dv1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Dvr+1P = 0 (where for consistency P ≡ 0 is a polynomial of degree −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Given a subgroup Γ3 < Γ1, we say that P : Γ1 → Γ2 is a polynomial map of degree at most r with respect to Γ3 if ∀ v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vr+1 ∈ Γ3 : Dv1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Dvr+1P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The following basic facts about polynomials will be useful for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2 below is due to Leibman [Lei02, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We include a short proof for the reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let P : Zd → R be a polynomial map with respect to a finite index subgroup L ≤ Zd, which is bounded, then P is constant on cosets of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' PERIODICITY OF JOINT CO-TILES IN Zd 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let r ∈ N denote the degree of P, as a polynomial with respect to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' It is clear from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 that if r is equal to 0, then the restriction of P to each coset of L is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Similarly, if the degree of P is equal to 1, then the restriction of P to each coset of L is a constant plus a non-trivial homomorphism (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' [Lei02]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For contradiction, we may assume that r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Observe that since P is bounded, for every v ∈ L we have DvP ⊆ P(Zd) − P(Zd), thus DvP is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Therefore, for every v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vr−1 ∈ L the function Dv1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Dvr−1P is a bounded polynomial map of degree exactly one, with respect to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' But non-trivial homomorphisms into R are unbounded, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' □ Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We say that a bounded function f : Zd → R has mean m if lim n→∞ 1 |Bn| � v∈Bn f(v) = m, (3) where Bn = {−n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , n}d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We say that f : Zd → R/Z is equidistributed in R/Z if lim n→∞ 1 |Bn| � v∈Bn g(f(v)) = � 1 0 g(x)dx (4) holds for every continuous function g : R/Z → R, where we identify g : R/Z → R with g : R → R such that g(x + 1) = x for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We will use the following version of Weyl’s equidistribution theorem for multivariate polynomials, see for instance [Yif22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='4 (Weyl’s equidistribution theorem for polynomials in several variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let P : Zd → R/Z be a polynomial map with respect to a finite index subgroup Γ of Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then on every coset v + Γ of Γ, the restriction of P to v + Γ is either equidistributed in R/Z or periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We implicitly rely on the following basic observation: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let F ⋐ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Suppose that F ⊂ Bn0 for some n0 ∈ N and that f : Zd → R is a bounded function satisfying 1F ∗ f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Denote by C = |F|(max f − min f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then for every n > n0 one has |Bn−n0| − C |Bn+n0 \\ Bn−n0| ≤ |F| � w∈Bn f(w) ≤ |Bn−n0| + C |Bn+n0 \\ Bn−n0| , (5) and thus the function f has mean 1 |F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In particular, if F1, F2 ⋐ Zd satisfy 1F1 ∗f = 1F2 ∗f = 1, then |F1| = |F2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Pick n0 ∈ N such that F ⊂ Bn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Observe that 1F ∗f = 1 implies that for every n > n0 we have 1Bn−n0 − C · 1Bn+n0\\Bn−n0 ≤ 1F ∗ f|Bn ≤ 1Bn−n0 + C · 1Bn+n0\\Bn−n0, where f|Bn denotes the restriction of f to Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Taking the sum of the values of these functions over all z ∈ Zd implies that (5) holds for every n > n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since limn→∞ |Bn−n0| |Bn| = 1 and limn→∞ |Bn+n0\\Bn−n0| |Bn| = 0, dividing (5) by |F| · |Bn| and letting n → ∞ yields the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' □ 6 TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The mean of a function f : Γ → R is defined similarly, using (3), for any countable amenable group Γ, in which case Bn is replaced by a Følner sequence in Γ, and an analogue of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5 holds in this more general context as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In Section 8, we implicitly apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5 for countable abelian groups Γ, which are in particular amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Shifts of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The space of co-tiles for a given finite set F ⊂ Zd, or more generally, the space of joint co-tiles for a given collection of sets, can naturally be endued with the structure of a compact topological space on which Zd acts by homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Topological dynamical systems of this kind are called Zd-subshifts, more specifically subshifts of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We include here relevant terminology and basic facts from the field of symbolic dynamics, particularly regarding shifts of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We refer to [LM95] for a comprehensive introduction to symbolic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let Σ be a finite set (alphabet) and Γ a finitely generated abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The set of functions from Γ to Σ, denoted ΣΓ, is called the full Γ-shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For x ∈ ΣΓ and v ∈ Γ, we use xv to denote the value of x at v (this is an element of Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Also for x ∈ ΣΓ and v ∈ Γ we denote by σv(x) ∈ ΣΓ the shift of x by v, which is given by σv(x)w = xv+w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Endowing ΣΓ with the product topology, where the topology on Σ is the discrete topology, makes ΣΓ a compact Γ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' A closed, non-empty and Γ-invariant subset X ⊆ ΣΓ is called a Γ-subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For x ∈ ΣΓ, the stabilizer of x is defined to be stab(x) = {v ∈ Γ : σv(x) = x}, which is a (possibly trivial) subgroup of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' A point x ∈ ΣΓ is called k-periodic if stab(x) is a subgroup of rank k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' When Γ = Z, we say that x ∈ ΣZ is periodic if it has a non-trivial stabilizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' A Γ-subshift X ⊆ ΣΓ is called a subshift of finite type (SFT) if there exists a finite set W ⊂ Γ and a set F ⊆ ΣW such that X = � x ∈ ΣΓ : ∀v ∈ Γ, σv(x)|W ̸∈ F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For every F ⋐ Zd the space of co-tiles for F is a subshift of finite type, under the natural identification of the space of co-tiles for F with XF := � x ∈ {0, 1}Zd : 1F ∗ x = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' To see that XF is indeed an SFT, take W = −F and F = � p ∈ {0, 1}W : � w∈W p(w) ̸= 1 � , and then XF = � x ∈ {0, 1}Zd : ∀v ∈ Zd, σv(x)|W ̸∈ F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since a non-empty intersection of SFTs is also an SFT, it follows that the space of joint co-tiles for a collection of tiles is an SFT (unless it is empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The following simple result is based on a pigeonhole argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The proof is well-known and standard, we include it for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Every Z-subshift of finite type admits a periodic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' PERIODICITY OF JOINT CO-TILES IN Zd 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let X ⊆ ΣZ be a Z-subshift of finite type, where Σ is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then by definition, there exists a finite set W ⋐ Z and F ⊆ ΣW such that X = � x ∈ ΣZ : ∀v ∈ Z, σv(x)|W ̸∈ F � , and X ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Fix x ∈ X, and let N ∈ N be an integer bigger than max(W) − min(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since the set Σ{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',N} is finite, by the pigeonhole principle there exist integers 0 ≤ i < j ≤ |Σ|N such that x|{i,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',i+N−1} = x|{j,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',j+N−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let p = j − i and define ˆx ∈ ΣZ by ˆxn = xi+(n mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then ˆx is a periodic point, and for every n ∈ Z there exists t ∈ {i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , j − 1} such that ˆx|W+n = x|W+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Hence, ˆx ∈ X, which proves that X admits a periodic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' □ We recall the following result in multidimensional symbolic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let Γ be a finitely generated abelian group, Γ0 ≤ Γ a subgroup, and X ⊆ ΣΓ a Γ-subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let XΓ0 := {x ∈ X : Γ0 ≤ stab(x)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (6) If XΓ0 ̸= ∅ then it is a Γ-subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Furthermore, if X is a subshift of finite type then XΓ0 is also a subshift of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' First, we show that XΓ0 is a subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since Γ is abelian, for every v ∈ Γ, v0 ∈ Γ0 and y ∈ XΓ0 we have σv0(σv(y)) = σv(σv0(y)) = σv(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' This shows σv(y) ∈ XΓ0 for all v ∈ Γ hence XΓ0 is Γ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' To see that XΓ0 is a closed subset of ΣΓ, consider a sequence (yn)n∈N ∈ XΓ0 such that lim n→∞ yn = y ∈ ΣΓ in the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since each yn ∈ XΓ0 ⊆ X and X is a closed subset of ΣΓ, we get y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Note that for any v0 ∈ Γ0, σv0(y) = σv0 � lim n→∞yn � = lim n→∞(σv0(yn)) = lim n→∞(yn) = y, which shows y ∈ XΓ0 and hence XΓ0 is a subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Now assuming that X is an SFT we show that XΓ0 is also an SFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Observe that XΓ0 = X ∩ Y where Y = {x ∈ ΣΓ : Γ0 ≤ stab(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since Γ0 is a subgroup of a finitely generated abelian group it is also finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let {γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , γr} be a finite generating set for Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then Y = r� i=1 {x ∈ ΣΓ : ∀v ∈ Γ, xv+γi = xv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' To see that Y is an SFT, let W = {0, γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , γr} and F = � w ∈ ΣW : ∃1 ≤ i ≤ r s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' w0 ̸= wγi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then Y = � x ∈ ΣΓ : ∀v ∈ Γ, σv(x)|W /∈ F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Hence Y is an SFT, which completes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 8 TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON □ From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='9 we deduce the following: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let Γ be a finitely generated abelian group of rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' If X ⊆ ΣΓ is a Γ-subshift of finite type that admits a (d − 1)-periodic point then it admits a d-periodic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Suppose X ⊆ ΣΓ is a Γ-subshift of finite type that admits a (d − 1)-periodic point, namely a point z ∈ X and a subgroup Γ0 ≤ Γ of rank d − 1 such that stab(z) = Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let XΓ0 be given by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then XΓ0 is non-empty, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='9 it is a subshift of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Because rank(Γ0) = d − 1, it follows that rank(Γ/Γ0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let v ∈ Zd be a vector such that k · v ̸∈ Γ0 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then Γ0 ⊕ Zv is a finite index subgroup of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let D ⊆ Γ be a fundamental domain for Γ0 ⊕ Zv, namely a finite set such that Γ0 ⊕ Zv ⊕ D = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Because D ⊕ Zv is a fundamental domain for Γ0 in Γ, it follows that the restriction map ρ : XΓ0 → ΣD⊕Zv is injective, where ρ is given by ρ(x) = x |D⊕Zv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Indeed, the inverse ρ−1 : ρ(XΓ0) → XΓ0 is given by ρ−1(˜x)u = (˜x)u′ for u ∈ Γ, where u′ is is the unique element in (D ⊕ Zv) that satisfies u − u′ ∈ Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Using the natural identification ΣD⊕Zv ∼= (ΣD)Z, we can view ρ(XΓ0) as a subset of (ΣD)Z, which we denote by ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let us show that ˜X is a Z-subshift of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Because X is a Γ-subshift of finite type, there exists a finite set W ⊂ Γ and F ⊂ ΣW such that XΓ0 is equal to the set of x ∈ ΣΓ satisfying σv(x) = x and σv(x) |W̸∈ F for all v ∈ Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We can assume without loss of generality that W is a subset of Zv ⊕ D, because Zv ⊕ D is a fundmental domain for Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let ˜W = {n ∈ Z : (nv + D) ∩ W ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then W = � n ˜ W(W ∩ (nv + D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Thus, there is a natural bijection between ΣW and (ΣD) ˜ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let ˜F denote the image of F under this bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then it follows directly that ˜X = � x ∈ (ΣD)Z : ∀v ∈ Z : σv(x) | ˜ W̸∈ ˜F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' This proves that ˜X is indeed a Z-subshift of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since ˜X is a Z-subshift of finite type, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='8 there exists a periodic point ˜z in ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let x = ρ−1(˜z), then x ∈ X is a d-periodic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The periodic decomposition theorem The following theorem asserts a certain decomposition for a joint co-tile of k-tuple of tiles in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The case where k = 1 and f is {0, 1}-valued essentially coincides with [GT21a, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='7], which is closely related to [Bha20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In the particular case that the tuple of tiles is independent, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 is a direct consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Namely, the indicator function of any joint co-tile of k independent tiles is a sum of k-periodic functions, each taking values in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The goal of this section is to prove the periodic decomposition theorem for joint co-tiles and to deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 (Periodic decomposition theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk ⋐ Zd, with 0 ∈ Fi for all 1 ≤ i ≤ k, and let f : Zd → Z be a bounded function that satisfies 1Fi ∗f = 1 for all 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We denote by S := |F1| = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' = |Fk| (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then for every 1 ≤ i ≤ k and every (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i there exists a function φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi : Zd → [min f, max f] with the following properties: PERIODICITY OF JOINT CO-TILES IN Zd 9 (a) For i < k we have φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi = 1 − � vi+1∈F ∗ i+1 φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi,vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (b) f = (−1)i � (v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi)∈F ∗ 1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='×F ∗ i φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi + i � j=1 (−(S − 1))j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (c) Let q denote the product of all primes less than or equal to (max f − min f)S, then (Zqv1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' + Zqvi) ≤ stab(φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi), (d) 1Fj ∗ φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi = 1 for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In particular, φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi has mean 1/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' There are various extensions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Some of these generalizations have further applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For the sake of readability, we do not state the most general form and instead indicate certain generalizations in the following sections, at the expense of some repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 relies on Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Various versions of this lemma, which is referred to as the dilation lemma, have been proved in [GT21a, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1], [Bha20, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1] for Γ = Zd, d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We also refer our readers to [Tij95, Theorem 1] where this lemma is proved for integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The proof is based on some elementary commutative algebra and it easily extends to countable abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For the sake of self-containment, we include a sketch of the proof below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The proof below is nearly identical to [GT21a, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1], except that we apply the assumption that r is co-prime to the order of torsion elements directly before eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2 (Dilation lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let Γ be a countable abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let 0 ∈ F ⋐ Γ, ℓ ∈ N and f : Γ → Z a bounded function satisfying 1F ∗ f = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let q1 be the product of all primes less than or equal to (max f − min f)|F|, let q2 be the product of all the orders of the torsion elements in (F − F), and set q = q1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then 1rF ∗ f = ℓ, for all r ∈ N such that r = 1 mod q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We use the notation f ∗p = f ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' ∗ f � �� � ×p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For any prime p we have 1∗p F = �� v∈F δv �∗p = � v∈F δ∗p v mod p, where the last equality holds by the Frobenius identity (f + g)∗p = f ∗p + g∗p mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For integers p that are co-prime to q2 we have that p(v1 − v2) ̸= 0 for any v1 ̸= v2 ∈ F, so the function v �→ pv is injective on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Thus: � v∈F δ∗p v = � v∈F δpv = 1pF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (7) 10 TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON Now convolving both sides of 1F ∗ f = ℓ by 1∗(p−1) F yields 1∗p F ∗ f = ℓ|F|p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Combining the above, for primes p that are co-prime to q2 we obtain 1pF ∗ f = ℓ|F|p−1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' If additionally p is co-prime to |F| by Fermat little theorem |F|p−1 = 1 mod p, thus 1pF ∗ f = ℓ mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Note that both 1F ∗f and 1pF ∗f take values in [|F| min f, |F| max f].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Recall that ℓ = 1F ∗f, so ℓ ∈ [|F| min f, |F| max f].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Thus, for p that is also greater than the size of that interval, the above equality holds without the mod p, namely 1pF ∗ f = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Finally, for r = 1 mod q, r is a product of primes that satisfy the conditions above, and the result follows by iterating the equation 1pF ∗ f = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' □ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For 1 ≤ i ≤ k, (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i and N ∈ N denote: φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi := 1 N i N � n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',ni=1 δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='+(1+niq)vi ∗ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (8) Let q be the product of all primes less than or equal to (max f − min f)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' By applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2 for Fj with Γ = Zd and ℓ = 1 we get 1rFj ∗ f = 1 for every r ∈ qN + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since 0 ∈ Fj we obtain f = 1 − � v∈F ∗ j δrv ∗ f for every 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For every N ∈ N, setting r = 1 + nq for n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , N} and taking average we conclude that for every 1 ≤ j ≤ k we have f = 1 − � v∈F ∗ j 1 N N � n=1 δ(1+nq)v ∗ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (9) Since φ(N) v1 = 1 N �N n=1 δ(1+nq)v1 ∗ f this gives (with j = 1): f = 1 − � v1∈F ∗ 1 φ(N) v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (10) For 1 ≤ i < k, choose any (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i and 1 ≤ n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , ni ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Setting j = i + 1 in (9) and convolving both sides of the equation by δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='+(1+niq)vi we obtain δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='+(1+niq)vi ∗ f = 1 − � vi+1∈F ∗ i+1 1 N N � ni+1=1 δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='+(1+niq)vi+(1+ni+1q)vi+1 ∗ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' By averaging over 1 ≤ n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , ni ≤ N and applying the definition in (8) we obtain that φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi = 1 − � vi+1∈F ∗ i+1 1 N i+1 N � n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',ni+1=1 δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='+(1+ni+1q)vi+1 ∗ f = 1 − � vi+1∈F ∗ i+1 φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (11) Since |F ∗ i | = S − 1 for 1 ≤ i ≤ k, using (10), (11) and an inductive argument we obtain that for every N ∈ N and 1 ≤ i ≤ k we have f = i � j=1 (−(S − 1))j−1 + (−1)i � (v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi)∈F ∗ 1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='×F ∗ i φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi (12) PERIODICITY OF JOINT CO-TILES IN Zd 11 Notice that the functions δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='+(1+niq)vi ∗ f are bounded between min f and max f, thus by (8), the functions φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi are bounded between min f and max f for every (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In particular, for every (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i the sequence of functions (φN v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi)N∈N is uniformly bounded, hence by Arzel`a–Ascoli theorem (or by a Cantor diagonal- ization argument), it converges along a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We denote the limit by φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then for every (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i we have min f ≤ φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi ≤ max f, and in view of (11) and (12) we have achieved (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' To see (c), using (8), a standard telescoping argument shows that for every w ∈ Zd, v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i and every 1 ≤ j ≤ i we have ��φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi(w + qvj) − φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi(w) �� ≤ 2N k−1 N k = 2 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Thus for every (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i the function φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi is qvj-periodic for every 1 ≤ j ≤ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' It is left to see (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Clearly, since 1Fj ∗ f = 1, for every 1 ≤ i, j ≤ k, (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i and n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , ni ∈ N we have 1Fj ∗ (δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='(1+niq)vi ∗ f) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Thus, by (8), 1Fj ∗ φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi = 1 for every N ∈ N and therefore 1Fj ∗ φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi = 1 for every 1 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In particular, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5, φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi has mean 1/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Under the assumption that f is {0, 1}-valued, it directly follows from The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1, part (a), that for every 1 ≤ i < k and every (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i , the sum � vi+1∈F ∗ i+1 φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi,vi+1 is a [0, 1]-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1, where k = 1 and f is {0, 1}-valued, coincides with [GT21a, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We will not make use of the property that 1Fj ∗ φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi = 1 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We mention it only for completeness and possibly for future reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The fact that the functions φv1 each have mean 1/S played an implicit role in [Bha20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Using the assumption that the tuple of tiles is independent Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 is an immediate corollary of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1, with f being a {0, 1}-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Suppose that (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd) is an independent tuple of tiles in Zd and that f : Zd → Z is a bounded function satisfying 1Fi ∗ f = 1 for all 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5, we have |F1| = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' = |Fd| := S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let q be the product of all primes less than or equal to (max f − min f)S and let L = � (v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vd)∈F ∗ 1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='×F ∗ d qZv1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' + qZvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 with k = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' It follows that f is a sum of functions whose stabilizers are rank d-subgroups, more precisely, f = (−1)d � (v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vd)∈F ∗ 1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='×F ∗ d φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vd + d � j=1 (−(S − 1))j−1, and for each (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vd) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ d we have that qZv1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' + qZvd ≤ stab(φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' By the above, stab(f) contains the intersection of stab(φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vd) over (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vd) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ d , that in turn contains L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 12 TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON By the assumption that the tuple (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fd) is independent, qZv1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' + qZvd is a finite index subgroup of Zd for every (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vd) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since L is an intersection of finitely many finite index subgroups, L is also a finite index subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' This proves that f is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Joint co-tilings in finitely generated abelian groups It is natural to ask which of the results about tilings generalize from Zd to more general groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' An inspection of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 reveals that the statement still holds, and the same proof applies, if we replace Zd by an arbitrary countable abelian group Γ, and change the value of q in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 (c) by multiplying it with the product of the orders of all torsion elements in F − F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' There is a simple observation that allows one to reduce statements about tilings of countable abelian groups by a finite set to the finitely generated case: Let Γ be a countable abelian group and let F ⋐ Γ with 0 ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let Γ0 denote the group generated by the difference set F − F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The assumption 0 ∈ F implies that F ⋐ Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then for any co-tile A of F we have that A ∩ Γ0 is a co-tile of F in Γ0, and tilings of Γ by F decompose into tilings of cosets of Γ0 in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' A corresponding statement is true also for a tuple of tiles (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk) and a joint co-tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Recall that g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , gk in a countable abelian group Γ are called independent if the equation �k j=1 njgj = 0, with n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , nk ∈ Z, implies that n1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' = nk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' With this definition, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 extends directly as follows: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let Γ be a countable abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For every k ∈ N the indicator function of any joint co-tile for k independent tiles in Γ is equal, up to a constant, to a sum of [0, 1]-valued functions whose stabilizer has rank at least k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Similarly, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2 extends as follows: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let Γ be a finitely generated abelian group of rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Any joint co-tile for d independent tiles in Γ has a finite orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' A quick remark about the condition of independence for a tuple of tiles for finitely generated abelian groups with non-trivial torsion: If Γ is of the form Γ = Zd × G where G is a finite abelian group and (F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk) is an independent tuple of tiles in Γ, then the only torsion element in each of the sets Fi is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For this reason, Newman’s theorem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' any tiling of Z by a finite set is periodic) does not hold in abelian groups Γ that are finite extensions of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Indeed, take Γ = Z × G, where G is a finite abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Take F = {1} × G ⋐ Γ, then the co-tiles of F are all the sets A ⊂ Γ of the following form: A = {(n, gn) : n ∈ Z}, for some sequence (gn)n∈Z of elements in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In particular, it is no longer true that any co-tile of F must be periodic, unless G is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Nonetheless, if G is a finite cyclic group of prime order, then the only obstructions to extending Newman’s theorem are of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' If Γ = Z × (Z/pZ) for some prime number p and F ⋐ Γ is a finite set, then every co-tile of F is periodic, unless F is of the form F = ˜F × (Z/pZ) for some finite tile ˜F ⋐ Z, in which case the co-tiles of F are all of the form A = {(n, gn) : n ∈ ˜A}, gn ∈ Z/pZ, (13) where ˜A is a co-tile of ˜F ⋐ Z, which by Newman’s theorem must be periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' PERIODICITY OF JOINT CO-TILES IN Zd 13 The proof of the proposition relies on the following generalization of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let Γ be a countable abelian group, F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , Fk ⋐ Γ such that |Fi| = S, and 0 ∈ Fi for all 1 ≤ i ≤ k, and let f : Γ → Z be a bounded function that satisfies 1Fi ∗ f = 1 for all 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For every 1 ≤ i ≤ k, let F Tor i denote the intersection of Fi with the torsion subgroup of Γ, and let F ∗ i = Fi \\ F Tor i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then for every 1 ≤ i ≤ k and every (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , vi) ∈ F ∗ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' × F ∗ i there exists a function φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi : Γ → [min f, max f] with the following properties: (a) For i < k we have 1F Tor i+1 ∗ φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi = 1 − � vi+1∈F ∗ i+1 φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi,vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (b) For every 1 ≤ i ≤ k there is an integer constant Ci such that 1F Tor 1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' ∗ 1F Tor i ∗ f = (−1)i � (v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi)∈F ∗ 1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='×F ∗ i φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi + Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (c) Let q1 be the product of all primes less than or equal to (max f − min f)S, let q2 be the product of all the orders of the torsion elements in the sets Fi − Fi, for 1 ≤ i ≤ k, and set q = q1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then (Zqv1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' + Zqvi) ≤ stab(φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi), (d) 1Fj ∗ φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi = 1 for all 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In particular, φv1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi has mean 1/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='4 below is a minor adaptation of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Note that in the case where Γ is a torsion free abelian group, F Tor i = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In particular, when Γ = Zd, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='4 coincides with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' By applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='2 for Fi with ℓ = 1 and q as in (c) we get 1rFi ∗ f = 1 for every r ∈ qN + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Because r = 1 mod q, we have rF Tor i = F Tor i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since Fi = F Tor i ⊎ F ∗ i we have 1F Tor i ∗ f = 1 − � v∈F ∗ i δrv ∗ f for every 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For every N ∈ N, setting r = 1 + nq for n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , N} and taking average we conclude that for every 1 ≤ j ≤ k we have 1F Tor j ∗ f = 1 − � vj∈F ∗ j 1 N N � nj=1 δ(1+njq)vj ∗ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (14) Applying (14) with j = i + 1, convolving both sides by δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='+(1+niq)vi and taking average over 1 Ni �N n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',ni=1 yields 1F Tor i+1 ∗ � 1 N i N � n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',ni=1 δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='+(1+niq)vi ∗ f � = 1 − � vi+1∈F ∗ i+1 1 N i+1 N � n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',ni,ni+1=1 δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='+(1+niq)vi+(1+ni+1q)vi+1 ∗ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 14 TOM MEYEROVITCH, SHREY SANADHYA, AND YAAR SOLOMON Defining φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi = 1 Ni �N n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',ni=1 δ(1+n1q)v1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='+(1+niq)vi ∗ f, as in (8), we obtain 1F Tor i+1 ∗ φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi = 1 − � vi+1∈F ∗ i+1 φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi,vi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (15) Note that (14) with j = 1 becomes 1F Tor 1 ∗ f = 1 − � v1∈F ∗ 1 φ(N) v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Convolving both sides by 1F Tor 2 and using (15) with i = 1 gives 1F Tor 1 ∗ 1F Tor 2 ∗ f = |F Tor 2 | − � v1∈F ∗ 1 1F Tor 2 ∗ φ(N) v1 = |F Tor 2 | − � v1∈F ∗ 1 � �1 − � v2∈F ∗ 2 φ(N) v1,v2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' By an inductive argument we obtain that for every N ∈ N and 1 ≤ i ≤ k there is a constant Ci ∈ Z, that does not depend on N, such that 1F Tor 1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' 1F Tor i ∗ f = Ci + (−1)i � (v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi)∈F ∗ 1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='×F ∗ i φ(N) v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=',vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' (16) Items (a) and (b) follow from (15) and (16) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The rest of the proof is completely identical to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 and therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let p be a prime number and let ∅ ̸= F0 ⫋ Z/pZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then 1F0 is an invertible element of the ring QZ/pZ, where multiplication in the ring is convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In other words, there exists g ∈ QZ/pZ such that g ∗ 1F0 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Consider the ring Q[x]/⟨xp − 1⟩ (with operations of addition and multiplication of polynomials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' It is easy to check that this ring is isomorphic as a ring to QZ/pZ, with the operations of pointwise addition and convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The isomorphism is given by identifying an element p−1 � i=0 aixi + ⟨xp − 1⟩ ∈ Q[x]/⟨xp − 1⟩ with the function f ∈ QZ/pZ given by f(i + pZ) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let F0 ⊂ Z/pZ be a non-empty proper subset of Z/pZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then 1F0 ∈ QZ/pZ is naturally identified with the coset of the polynomial P(x) = � (i+pZ)∈F0 xi in Q[x]/⟨xp − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then the assumption that F0 is a non-empty proper subset of Z/pZ implies that the polynomial P is co-prime to the cyclotomic polynomial of order p, Φp = �p−1 i=0 xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since P(1) = |F0| ̸= 0 it follows that P is co-prime to x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Because xp − 1 = Φp(x)(x − 1), it follows that P is co-prime to xp − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Hence there exists polynomials Q1, Q2 ∈ Q[x] such that 1 = Q1(x)P(x) + Q2(x)(xp − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' This means that in the ring Q[x]/⟨xp − 1⟩, the coset of Q1(x)P(x) is the same as the coset of the polynomial 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since the coset of the polynomial 1 in Q[x]/⟨xp − 1⟩ corresponds to δ0 ∈ QZ/pZ, this implies that g ∗ 1F0 = δ0, where g ∈ QZ/pZ is the element corresponding to the coset of Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let p be a prime number and F ⋐ Z × (Z/pZ) be a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Suppose A ⊂ Z × (Z/pZ) satisfies 1F ∗ 1A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='4 with Γ = Z × (Z/pZ) k = 1, F1 = F and f = 1A, we conclude that 1F Tor ∗ 1A is a sum functions having infinite stabilizer, hence 1F Tor ∗ 1A is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' PERIODICITY OF JOINT CO-TILES IN Zd 15 First, assume that there is a set ˜F ⋐ Z such that F = ˜F×Z/pZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' So 1F = 1 ˜F×{0}∗1{0}×(Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Thus 1 ˜F×{0} ∗ 1{0}×(Z/pZ) ∗ 1A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' This implies that 1{0}×(Z×pZ) ∗ 1A ≤ 1, so for every n ∈ Z there exists at most one element gn ∈ Z/pZ such that (n, gn) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Hence, in this case, A is of the form (13) for some set ˜A ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' It follows that 1 ˜F ∗ 1 ˜ A = 1, where the convolution here is with respect to the group Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Now suppose that F is not of the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' This means that there exists n ∈ Z such that F ∩ ({n} × Z/pZ) is a non-empty proper subset of {n} × (Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' By translating F we can assume without loss of generality that F Tor is neither empty nor equal to {0} × (Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then there exists a non-empty proper subset F0 ⊂ Z/pZ such that F Tor = {0} × F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' In this case, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5, there exists g : Z/pZ → Q such that g ∗ 1F0 = δ0, where the convolution is in (Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Let ˜g : Z × Z/pZ → Q be given by ˜g(0, i) = g(i) for i ∈ Z/pZ and g(n, i) = 0 for every n ∈ Z \\ {0} and i ∈ Z/pZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Then ˜g ∗ 1F Tor = δ0, where this time the convolution is in Z × (Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Since 1F Tor ∗ 1A is periodic, so is ˜g ∗ 1F Tor ∗ 1A = 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We have thus shown that in the case that F is not of the form F = ˜F × (Z/pZ) for some ˜F ⋐ Z, every co-tile is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Property (⋆) implies (d − 1)-piecewise periodicity In this section, we use property (⋆) to deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' To this end, we will use Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='4, which is a version of Weyl’s equidistribution theorem for polynomials in several variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' The relevance of Weyl’s equidistribution theorem to our setting comes from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' We note that similar arguments have appeared earlier in [Bha20], [KS20] and [GT21a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Suppose g, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' , gm : Γ1 → Γ2 are functions, where Γ1, Γ2 are abelian groups, such that �m i=1 gi = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' Suppose g is a polynomial of degree at most r ∈ N with respect to a subgroup Γ0 ≤ Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFIT4oBgHgl3EQfaCs0/content/2301.11255v1.pdf'} +page_content=' For any 1 ≤ i < j ≤ m define the group Li,j = stab(gi) + stab(gj), and let L = � 1≤i