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+arXiv:2301.03277v1  [math.DG]  9 Jan 2023
+Functionals for the Study of LCK Metrics on Compact
+Complex Manifolds
+Dan Popovici and Erfan Soheil
+Abstract. We propose an approach to the existence problem for locally conformally K¨ahler metrics on
+compact complex manifolds by introducing and studying a functional that is different according to whether
+the complex dimension of the manifold is 2 or higher.
+1
+Introduction
+Let X be an n-dimensional compact complex manifold with n ≥ 2. In this paper, we propose a
+variational approach to the existence of locally conformally K¨ahler (lcK) metrics on X by introducing
+and analysing a functional in each of the cases n = 2 and n ≥ 3. This functional, defined on the
+non-empty set HX of all the Hermitian metrics on X, assumes non-negative values and vanishes
+precisely on the lcK metrics. We compute the first variation of our functional on both surfaces and
+higher-dimensional manifolds.
+We will identify a Hermitian metric on X with the associated C∞ positive definite (1, 1)-form
+ω. The set HX of all these metrics is a non-empty open convex cone in the infinite-dimensional
+real vector space C∞
+1, 1(X, R) of all the real-valued smooth (1, 1)-forms on X. As is well known,
+a Hermitian metric ω is called K¨ahler if dω = 0 and a complex manifold X is said to be K¨ahler
+if there exists a K¨ahler metric thereon. Meanwhile, the notion of locally conformally K¨ahler
+(lcK) manifold originates with I. Vaisman in [Vai76]. There are several equivalent definitions of
+lcK manifolds. The one adopted in this paper stipulates that a complex manifold X is lcK if there
+exists an lcK metric thereon, while a Hermitian metric ω on X is said to be lcK if there exists a C∞
+1-form θ on X such that dθ = 0 and
+dω = ω ∧ θ.
+When it exists, the 1-form θ is unique and is called the Lee form of ω. For equivalent definitions of
+lcK manifolds, the reader is referred e.g. to Definitions 3.18 and 3.29 of [OV22].
+One of the early results in the theory of lcK manifolds is Vaisman’s theorem according to which
+any lcK metric on a compact K¨ahler manifold is, in fact, globally conformally K¨ahler. This theorem
+was extended to compact complex spaces with singularities by Preda and Stanciu in [PS22].
+The question of when lcK metrics exist on a given compact complex manifold X has been
+extensively studied. For example, Otiman characterised the existence of such metrics with prescribed
+Lee form in terms of currents: given a d-closed 1-form θ on X and considering the associated twisted
+operator dθ = d+θ∧·, Theorem 2.1 in [Oti14] stipulates that X admits an lcK metric whose Lee form
+is θ if and only if there are no non-trivial positive (1, 1)-currents on X that are (1, 1)-components of
+dθ-boundaries.
+On the other hand, Istrati investigated the relation between the existence of special lcK metrics
+on a compact complex manifold and the group of biholomorphisms of the manifold. Specifically,
+according to Theorem 0.2 in [Ist19], a compact lcK manifold X admits a Vaisman metric if the
+group of biholomorphisms of X contains a torus T that is not purely real. A compact torus T of
+1
+
+biholomorphisms of a compact complex manifold (X, J) is said to be purely real (in the sense of
+(1) of Definition 0.1. in [Ist19]) if its Lie algebra t satisfies the condition t ∩ Jt = 0, where J is the
+complex structure of X. Recall that an lcK metric ω is said to be a Vaisman metric if ∇ωθ = 0,
+where θ is the Lee form of ω and ∇ω is the Levi-Civita connection determined by ω.
+The approach we propose in this paper to the issue of the existence of lcK metrics on a compact
+complex n-dimensional manifold X is analytic. Given an arbitrary Hermitian metric ω on X, the
+Lefschetz decomposition
+dω = (dω)prim + ω ∧ θω
+of dω into a uniquely determined ω-primitive part and a part divisible by ω with a uniquely de-
+termined quotient 1-form θω (the Lee form of ω) gives rise to the following dichotomy (cf. Lemma
+2.2):
+(i) either n = 2, in which case (dω)prim = 0 but the Lee form θω need not be d-closed, so the lcK
+condition on ω is equivalent to dθω = 0. This turns out to be equivalent to ∂θ1, 0
+ω
+= 0. Therefore, we
+define our functional L : HX −→ [0, +∞) in this case to be
+L(ω) = ||∂θ1, 0
+ω ||2
+ω,
+namely its value at every Hermitian metric ω on X is defined to be the squared L2
+ω-norm of ∂θ1, 0
+ω .
+(ii) or n ≥ 3, in which case the lcK condition on ω is equivalent to the vanishing condition
+(dω)prim = 0.
+This is further equivalent to the vanishing of either (∂ω)prim or (¯∂ω)prim.
+We,
+therefore, define our functional L : HX −→ [0, +∞) in this case to be
+L(ω) = ||(¯∂ω)prim||2
+ω,
+namely its value at every Hermitian metric ω on X is defined to be the squared L2
+ω-norm of the
+ω-primitive part of the (1, 2)-form ¯∂ω.
+The main results of the paper are the computations of the first variation of our functional L in
+each of the cases n = 2 (cf. Theorem 4.4) and n ≥ 3 (cf. Theorem 5.1).
+While the functional L is scaling-invariant when n = 2, this fails to be the case when n ≥ 3. In
+this latter case, we obtain two proofs – one as a corollary of the formula for the first variation of our
+functional (cf. Proposition 5.3), the other as a direct consequence of the behaviour of our functional
+in the scaling direction (cf. Proposition 6.2) – for the equivalence:
+ω is a critical point for the functional L if and only if ω is lcK
+Still in the case n ≥ 3, we introduce in Definition 6.5 a normalised version �Lρ of the functional L
+depending on an arbitrary background Hermitian metric ρ. The first variation of �Lρ is then deduced
+in Proposition 6.6 from the analogous computation for L obtained in Theorem 5.1. One motivation
+for the normalisation we propose in terms of a (possibly balanced and possibly moving) metric ρ
+stems from the conjecture predicting that the simultaneous existence of a balanced metric and of
+an lcK metric on a compact complex manifold ought to imply the existence of a K¨ahler metric. We
+hope to be able to develop this line of thought in future work.
+2
+
+At the end of §.6, we use our scaling-invariant functionals L (in the case of compact complex
+surfaces) and �Lρ (in the case of higher-dimensional compact complex manifolds) to produce positive
+(1, 1)-currents whose failure to be either C∞ forms or strictly positive provides possible obstructions
+to the existence of lcK metrics.
+Acknowledgments. This work is part of the second-named author’s thesis under the supervision
+of the first-named author. The former wishes to thank the latter for constant support.
+2
+Preliminaries
+In this section, we recast some standard material in the language of primitive forms and make a few
+observations that will be used in the next sections.
+Let X be a complex manifold with dimCX = n. We will denote by:
+(i) C∞
+k (X, C), resp. C∞
+p, q(X, C), the space of C∞ differential forms of degree k, resp. of bidegree
+(p, q) on X. When these forms α are real (in the sense that α = α), the corresponding spaces will
+be denoted by C∞
+k (X, R), resp. C∞
+p, q(X, R).
+(ii) ΛkT ⋆X, resp. Λp, qT ⋆X, the vector bundle of differential forms of degree k, resp. of bidegree
+(p, q), as well as the spaces of such forms considered in a pointwise way.
+For any (1, 1)-form ρ ≥ 0, we will also use the following notation:
+ρk := ρk
+k! ,
+1 ≤ k ≤ n.
+When ρ = ω is C∞ and positive definite (i.e. ω is a Hermitian metric on X), it can immediately be
+checked that
+dωk = ωk−1 ∧ dω
+and
+⋆ω ωk = ωn−k
+for all 1 ≤ k ≤ n, where ⋆ = ⋆ω is the Hodge star operator induced by ω.
+Recall the following standard
+Definition 2.1 A C∞ positive definite (1, 1)-form (i.e. a Hermitian metric) ω on a complex man-
+ifold X is said to be locally conformally K¨ahler (lcK) if
+dω = ω ∧ θ
+for some C∞ 1-form θ satisfying dθ = 0.
+The 1-form θ is uniquely determined, is real and is called the Lee form of ω.
+The obstruction to a given Hermitian metric ω being lcK depends on whether n = 2 or n ≥ 3.
+Lemma 2.2 Let X be a complex manifold with dimCX = n.
+(i) If n = 2, for any Hermitian metric ω there exists a unique, possibly non-closed, C∞ 1-form
+θ = θω such that dω = ω ∧ θ. Therefore, ω is lcK if and only if θω is d-closed.
+3
+
+Moreover, for any Hermitian metric ω, the 2-form dθω is ω-primitive, i.e. Λω(dθω) = 0, or
+equivalently, ω ∧ dθω = 0, while the Lee form is real and is explicitly given by the formula:
+θω = Λω(dω).
+(1)
+Alternatively, if θω = θ1, 0
+ω
++ θ0, 1
+ω
+is the splitting of θω into components of pure types, we have
+θ1, 0
+ω
+= Λω(∂ω) = −i¯∂⋆ω
+(2)
+and the analogous formulae for θ0, 1
+ω
+= θ1, 0
+ω
+obtained by taking conjugates.
+(ii) If n ≥ 3, for any Hermitian metric ω there exists a unique ω-primitive C∞ 3-form (dω)prim and
+a unique C∞ 1-form θ = θω such that dω = (dω)prim + ω ∧ θ. The Lee form is real and is explicitly
+given by the formula
+θω =
+1
+n − 1 Λω(dω).
+(3)
+Moreover, ω is lcK if and only if (dω)prim = 0.
+If ω is lcK, then
+θ1, 0
+ω
+=
+1
+n − 1 Λω(∂ω) = −
+i
+n − 1
+¯∂⋆ω
+(4)
+and the analogous formulae obtained by taking conjugates hold for θ0, 1
+ω
+= θ1, 0
+ω .
+Recall that for any k ≤ n and any Hermitian metric ω on X, the multiplication map
+Ll
+ω = ωl ∧ · : ΛkT ⋆X −→ Λk+2lT ⋆X
+defined at every point of X is an isomorphism if l = n−k, is injective (but in general not surjective)
+for every l < n − k and is surjective (but in general not injective) for every l > n − k. A k-form is
+said to be ω-primitive if it lies in the kernel of the multiplication map Ln−k+1
+ω
+. Equivalently, the
+ω-primitive k-forms are precisely those that lie in the kernel of Λω : ΛkT ⋆X −→ Λk−2T ⋆X.
+Also recall that for every k ≤ n, every k-form α admits a unique ⟨ , ⟩ω-orthogonal pointwise
+splitting (called the Lefschetz decomposition):
+α = αprim + ω ∧ β(1)
+prim + ω2 ∧ β(2)
+prim + · · · + ωr ∧ β(r)
+prim,
+(5)
+where r is the largest non-negative integer such that 2r ≤ k, αprim, β(1)
+prim, . . . , β(r)
+prim are ω-primitive
+forms of respective degrees k, k −2, . . . , k −2r ≥ 0, and ⟨ , ⟩ω is the pointwise inner product defined
+by ω. We will call αprim the primitive part of α.
+Finally, recall the Hermitian commutation relation:
+i[Λω, ∂] = −(¯∂⋆
+ω + ¯τ ⋆
+ω)
+(6)
+proved in [Dem84], where τω := [Λω, ∂ω ∧ ·] is the torsion operator of order 0 and bidegree (1, 0).
+This definition of τω yields
+¯τ ⋆
+ωω = [(¯∂ω ∧ ·)⋆, Lω](ω) = (¯∂ω ∧ ·)⋆(ω2).
+4
+
+On the other hand, if α1, 0 is any (1, 0)-form on X, let ¯ξα be the (0, 1)-vector field defined by
+the requirement ¯ξα⌟ω = α1, 0. It is easily checked in local coordinates chosen about a given point x
+such that the metric ω is defined by the identity matrix at x, that the adjoint w.r.t. ⟨ , ⟩ω of the
+contraction operator by ¯ξα is given by the formula
+(¯ξα⌟·)⋆ = −iα0, 1 ∧ ·,
+or equivalently
+− i¯ξα⌟· = (α0, 1 ∧ ·)⋆,
+where α0, 1 = α1, 0. Explicitly, if α0, 1 = �
+k
+¯akd¯zk on a neighbourhood of x, then −i¯ξα⌟· = (α0, 1∧·)⋆ =
+�
+k
+ak
+∂
+∂¯zk ⌟· at x. Hence, −i¯ξα⌟α0, 1 = �
+k
+|ak|2 = |α0, 1|2
+ω at x. We have just got the pointwise formula
+− i¯ξα⌟α0, 1 = |α0, 1|2
+ω = |α1, 0|2
+ω
+(7)
+at every point of X.
+Now, suppose that dω = ω ∧ θω for some (necessarily real) 1-form θω. Then, ¯∂ω = ω ∧ θ0, 1
+ω , so
+(¯∂ω ∧ ·)⋆ = −iΛω(¯ξθ⌟·), where ¯ξθ := ¯ξα with α1, 0 = θ1, 0
+ω . The above formula for ¯τ ⋆
+ωω translates to
+¯τ ⋆
+ωω = −iΛω(¯ξθ⌟ω2) = −2iΛω(ω ∧ (¯ξθ⌟ω)) = −2i[Λω, Lω](¯ξθ⌟ω) = −2i(n − 1)θ1, 0
+ω
+The conclusion of this discussion is that, when dω = ω ∧ θω, formula (3) translates to
+θ1, 0
+ω
+=
+1
+n − 1 Λω(∂ω) =
+1
+n − 1 [Λω, ∂](ω) =
+1
+n − 1 i¯∂⋆
+ωω +
+1
+n − 1 i¯τ ⋆
+ωω =
+1
+n − 1 i¯∂⋆
+ωω + 2θ1, 0
+ω ,
+which amounts to θ1, 0
+ω
+= −
+1
+n−1 i¯∂⋆
+ωω. This proves (4) for an arbitrary n, hence also (2) when n = 2,
+if the other statements in Lemma 2.2 have been proved.
+Proof of Lemma 2.2. (i) When n = 2, the map ω ∧ · : Λ1T ⋆X −→ Λ3T ⋆X is an isomorphism at
+every point of X. In particular, the 3-form dω is the image of a unique 1-form θ under this map.
+To see that dθ is primitive, we apply d to the identity dω = ω ∧ θ to get
+0 = d2ω = dω ∧ θ + ω ∧ dθ.
+Meanwhile, multiplying the same identity by θ, we get dω ∧ θ = ω ∧ θ ∧ θ = 0 since θ ∧ θ = 0 due
+to the degree of θ being 1. Therefore, ω ∧ dθ = 0, which means that the 2-form dθ is ω-primitive.
+To prove formula (1), we apply Λω to the identity dω = ω ∧ θ to get
+Λω(dω) = [Λω, Lω](θ) = −[Lω, Λω](θ) = −(1 − 2) θ = θ,
+where we used the identities Λω(θ) = 0 (for bidegree reasons) and [Lω, Λω] = (k − n) Id on k-forms
+(while here k = 1 and n = 2).
+(ii) The splitting dω = (dω)prim +ω ∧θ is the Lefschetz decomposition of dω w.r.t. the metric ω.
+Applying Λω, we get Λω(dω) = [Λω, Lω](θ) = −[Lω, Λω](θ) = −(1 − n) θ = (n − 1) θ, which proves
+(3).
+The implication “ω lcK =⇒ (dω)prim = 0“ follows at once from the definitions. To prove the
+reverse implication, suppose that (dω)prim = 0. We have to show that θ is d-closed. The assumption
+means that dω = ω ∧ θ, so dω ∧ θ = ω ∧ θ ∧ θ = 0 and 0 = d2ω = dω ∧ θ + ω ∧ dθ. Consequently,
+ω ∧ dθ = 0. Now, the multiplication of k-forms by ωl is injective whenever l ≤ n − k. When n ≥ 3,
+5
+
+if we choose l = 1 and k = 2 we get that the multiplication of 2-forms by ω is injective. Hence, the
+identity ω ∧ dθ = 0 implies dθ = 0, so ω is lcK.
+□
+Another standard observation is that the Lefschetz decomposition transforms nicely, hence the
+lcK property is preserved, under conformal rescaling.
+Lemma 2.3 Let ω be an arbitrary Hermitian metric and let f be any smooth real-valued function
+on a compact complex n-dimensional manifold X.
+If dω = (dω)prim + ω ∧ θω is the Lefschetz
+decomposition of dω w.r.t. the metric ω (with the understanding that (dω)prim = 0 when n = 2),
+then
+d(efω) = ef(dω)prim + efω ∧ (θω + df)
+(8)
+is the Lefschetz decomposition of d(efω) w.r.t. the metric �ω := efω.
+Consequently, ω is lcK if and only if any conformal rescaling efω of ω is lcK, while the Lee form
+transforms as θef ω = θω + df. In particular, when the lcK metric ω varies in a fixed conformal class,
+the Lee form θω varies in a fixed De Rham 1-class {θω}DR ∈ H1(X, R) called the Lee De Rham
+class associated with the given conformal class. Moreover, the map ω �→ θω defines a bijection from
+the set of lcK metrics in a given conformal class to the set of elements of the corresponding Lee De
+Rham 1-class.
+Proof. Differentiating, we get d(efω) = efdω + efω ∧ df = ef(dω)prim + efω ∧ (θω + df). Meanwhile,
+it can immediately be checked that
+Λefω = e−fΛω,
+so ker Λefω = ker Λω. Thus, the ω-primitive forms coincide with the �ω-primitive forms. Since Λ�ω
+commutes with the multiplication by any real-valued function, ef(dω)prim is �ω-primitive, so (8) is
+the Lefschetz decompostion of d�ω w.r.t. �ω.
+□
+When X is compact, we know from [Gau77] that every Hermitian metric ω on X admits a (unique
+up to a positive multiplicative constant) conformal rescaling �ω := efω that is a Gauduchon metric.
+These metrics are defined (cf. [Gau77]) by the requirement that ∂ ¯∂�ωn−1 = 0, where n is the complex
+dimension of X. This fact, combined with Lemma 2.3, shows that no loss of generality is incurred
+in the study of the existence of lcK metrics on compact complex manifolds if we confine ourselves
+to Gauduchon metrics.
+We end this review of known material with the following characterisation (cf. [AD15, Lemma
+2.5]) of Gauduchon metrics on surfaces in terms of their Lee forms.
+Lemma 2.4 Let ω be a Hermitian metric on a complex surface X. The following equivalence holds:
+∂ ¯∂ω = 0
+(i.e. ω is a Gauduchon metric)
+⇐⇒
+¯∂⋆
+ωθ0, 1
+ω
+= 0,
+where θ0, 1
+ω
+is the component of type (0, 1) of the Lee form θω of ω.
+In particular, d⋆
+ωθω = 0 if ω is Gauduchon.
+6
+
+Proof. We give a proof different from the one in [AD15] by making use of the Hermitian commutation
+relations. By applying ∂ to the identity ¯∂ω = ω ∧ θ0, 1
+ω
+and using the identity ∂ω = ω ∧ θ1, 0
+ω , we get
+∂ ¯∂ω = ∂ω ∧ θ0, 1
+ω
++ ω ∧ ∂θ0, 1
+ω
+= ω ∧ (θ1, 0
+ω
+∧ θ0, 1
+ω
++ ∂θ0, 1
+ω ).
+Taking Λω, we get
+Λω(∂ ¯∂ω) = [Λω, Lω](θ1, 0
+ω
+∧ θ0, 1
+ω
++ ∂θ0, 1
+ω ) + ω ∧ Λω(θ1, 0
+ω
+∧ θ0, 1
+ω
++ ∂θ0, 1
+ω ) = Λω(θ1, 0
+ω
+∧ θ0, 1
+ω
++ ∂θ0, 1
+ω ) ω,
+where the second identity follows from [Λω, Lω] = −(2 − 2) Id = 0 on 2-forms on complex surfaces.
+Now, Λω(θ1, 0
+ω
+∧ θ0, 1
+ω
++ ∂θ0, 1
+ω ) is a function, so from the above identities we get the equivalences
+Λω(∂ ¯∂ω) = 0
+⇐⇒
+Λω(θ1, 0
+ω
+∧ θ0, 1
+ω
++ ∂θ0, 1
+ω ) = 0 ⇐⇒ θ1, 0
+ω
+∧ θ0, 1
+ω
++ ∂θ0, 1
+ω
+is ω-primitive
+⇐⇒
+ω ∧ (θ1, 0
+ω
+∧ θ0, 1
+ω
++ ∂θ0, 1
+ω ) = 0 ⇐⇒ ∂ ¯∂ω = 0.
+We remember the equivalence ∂ ¯∂ω = 0 ⇐⇒ Λω(θ1, 0
+ω
+∧ θ0, 1
+ω ) + Λω(∂θ0, 1
+ω ) = 0. Since Λω(iθ1, 0
+ω
+∧
+θ0, 1
+ω ) = |θ1, 0
+ω |2
+ω (immediate verification) and Λωθ0, 1
+ω
+= 0 (for bidegree reasons), we get the equivalence:
+∂ ¯∂ω = 0 ⇐⇒ |θ1, 0
+ω |2
+ω + i[Λω, ∂] θ0, 1
+ω
+= 0.
+The Hermitian commutation relation i[Λω, ∂] = −(¯∂⋆
+ω + ¯τ ⋆
+ω) (cf. (6), see [Dem84]) transforms the
+last equivalence into
+∂ ¯∂ω = 0 ⇐⇒ |θ1, 0
+ω |2
+ω − (¯∂⋆
+ωθ0, 1
+ω
++ ¯τ ⋆
+ωθ0, 1
+ω ) = 0.
+(9)
+On the other hand, ¯τ ⋆
+ω = [(¯∂ω ∧ ·)⋆, ω ∧ ·]. From this we get
+Formula 2.5 For any Hermitian metric ω on a complex surface, we have
+¯τ ⋆
+ωθ0, 1
+ω
+= |θ0, 1
+ω |2
+ω.
+Proof of Formula 2.5. Since (¯∂ω∧·)⋆θ0, 1
+ω
+= 0 for bidegree reasons, we get ¯τ ⋆
+ωθ0, 1
+ω
+= (¯∂ω∧·)⋆(ω∧θ0, 1
+ω ).
+Since ¯∂ω = ω ∧ θ0, 1
+ω , we have (¯∂ω ∧ ·)⋆ = −iΛω(¯ξθ⌟·) (see (7) and the discussion there below), where
+¯ξθ is the (0, 1)-vector field defined by the requirement ¯ξθ⌟ω = θ1, 0
+ω . Hence
+¯τ ⋆
+ωθ0, 1
+ω
+= −iΛω(θ1, 0
+ω
+∧ θ0, 1
+ω ) − iΛω[ω ∧ (¯ξθ⌟θ0, 1
+ω )].
+Since −i¯ξθ⌟θ0, 1
+ω
+= |θ0, 1
+ω |2
+ω (cf. (7)), we infer that
+¯τ ⋆
+ωθ0, 1
+ω
+= −Λω(iθ1, 0
+ω
+∧ θ0, 1
+ω ) + 2 |θ0, 1
+ω |2
+ω,
+since Λω(ω) = n = 2. Meanwhile, θ1, 0
+ω
+= θ0, 1
+ω , so we get Λω(iθ1, 0
+ω ∧θ0, 1
+ω ) = |θ1, 0
+ω |2
+ω = |θ0, 1
+ω |2
+ω (immediate
+verification in local coordinates). Formula 2.5 is now proved.
+□
+End of proof of Lemma 2.4. Formula 2.5 transforms equivalence (9) into
+∂ ¯∂ω = 0 ⇐⇒ (|θ1, 0
+ω |2
+ω − |θ0, 1
+ω |2
+ω) − ¯∂⋆
+ωθ0, 1
+ω
+= 0 ⇐⇒ ¯∂⋆
+ωθ0, 1
+ω
+= 0
+and we are done
+□
+7
+
+3
+An enerygy functional for the study of lcK metrics
+In what follows, we will restrict attention to the set
+HX := {ω ∈ C∞
+1, 1(X, R) | ω > 0}
+of all Hermitian metrics on X. This is a non-empty open cone in the infinite-dimensional vector
+space C∞
+1, 1(X, R) of all smooth real (1, 1)-forms on X. It will be called the Hermitian cone of X.
+Building on Lemma 2.2, we introduce the following energy functional. By || ||ω, respectively
+| |ω, we mean the L2-norm, respectively the pointwise norm, defined by ω.
+Definition 3.1 Let X be a compact complex manifold with dimCX = n.
+(i) If n = 2, let L : HX −→ [0, +∞) be defined by
+L(ω) :=
+�
+X
+∂θ1, 0
+ω
+∧ ¯∂θ0, 1
+ω
+= ||∂θ1, 0
+ω ||2
+ω,
+where θω is the Lee form of ω.
+(ii) If n ≥ 3, let L : HX −→ [0, +∞) be defined by
+L(ω) :=
+�
+X
+i(¯∂ω)prim ∧ (¯∂ω)prim ∧ ωn−3 = ||(¯∂ω)prim||2
+ω,
+where (¯∂ω)prim is the ω-primitive part of ¯∂ω in its Lefschetz decomposition (5).
+This definition is justified by the following observation.
+Lemma 3.2 In the setup of Definition 3.1, for every metric ω ∈ HX the following equivalence holds:
+ω
+is an lcK metric ⇐⇒ L(ω) = 0.
+Proof. • In the case n = 2, we know from (i) of Lemma 2.2 that ω is lcK if and only if dθω = 0.
+This condition is equivalent to L(ω) = 0, where we set
+L(ω) := ||dθω||2
+ω =
+�
+X
+dθω ∧ ⋆(d¯θω).
+We also know from (i) of Lemma 2.2 that dθω is ω-primitive, so we get
+0 = Λω(dθω) = Λω(∂θ1, 0
+ω ) + Λω(∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω ) + Λω(¯∂θ0, 1
+ω ) = Λω(∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω ),
+where the last identity follows from the previous one for bidegree reasons. We infer that the (1, 1)-
+form ∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω
+is ω-primitive. But so are ∂θ1, 0
+ω
+and ¯∂θ0, 1
+ω
+for bidegree reasons, so we can apply
+the following general formula (cf. e.g. [Voi02, Proposition 6.29, p. 150]) that holds for any primitive
+form v of arbitrary bidegree (p, q) on any complex n-dimensional manifold:
+⋆ v = (−1)k(k+1)/2 ip−q ωn−p−q ∧ v,
+where k := p + q,
+(10)
+8
+
+to get
+⋆(dθω) = ∂θ1, 0
+ω
+− (∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω ) + ¯∂θ0, 1
+ω . We infer that
+dθω ∧ ⋆(d¯θω)
+=
+[∂θ1, 0
+ω
++ (∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω ) + ¯∂θ0, 1
+ω ] ∧ [∂θ1, 0
+ω
+− (∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω ) + ¯∂θ0, 1
+ω ]
+=
+2 ∂θ1, 0
+ω
+∧ ¯∂θ0, 1
+ω
+− (∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω )2
+and finally that
+L(ω) = 2 L(ω) −
+�
+X
+(∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω )2.
+(11)
+On the other hand, the Stokes formula implies the first of the following identities
+0
+=
+�
+X
+dθω ∧ dθω =
+�
+X
+[∂θ1, 0
+ω
++ (∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω ) + ¯∂θ0, 1
+ω ] ∧ [∂θ1, 0
+ω
++ (∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω ) + ¯∂θ0, 1
+ω ]
+=
+2 L(ω) +
+�
+X
+(∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω )2.
+(12)
+We conclude from (11) and (12) that L(ω) = 0 if and only if L(ω). Thus, we have proved that
+ω is lcK if and only if L(ω) = 0, as claimed.
+The identity L(ω) = ||∂θ1, 0
+ω ||2
+ω follows at once from the general formula (10) applied to the prim-
+itive (2, 0)-form ∂θ1, 0
+ω . Indeed, ⋆∂θ1, 0
+ω
+= ∂θ1, 0
+ω , hence ∂θ1, 0
+ω
+∧ ¯∂θ0, 1
+ω
+= ∂θ1, 0
+ω
+∧ ⋆(∂θ1, 0
+ω ) = |∂θ1, 0
+ω |2
+ω dVω.
+• In the case n ≥ 3, we know from (ii) of Lemma 2.2 that ω is lcK if and only if (dω)prim = 0.
+Now, (dω)prim = (∂ω)prim + (¯∂ω)prim and the forms (∂ω)prim and (¯∂ω)prim are conjugate to each
+other and of different pure types ((2, 1), respectively (1, 2)), so the vanishing of (dω)prim is equivalent
+to the vanishing of (¯∂ω)prim.
+Meanwhile, the standard formula (10) applied to the primitive (2, 1)-form (¯∂ω)prim = (∂ω)prim
+spells:
+⋆ (¯∂ω)prim = i (¯∂ω)prim ∧ ωn−3.
+This proves the identity L(ω) = ||(¯∂ω)prim||2
+ω.
+Putting these pieces of information together, we get the following equivalences:
+ω
+lcK ⇐⇒ (dω)prim = 0 ⇐⇒ (¯∂ω)prim = 0 ⇐⇒ L(ω) = 0.
+The proof is complete.
+□
+4
+First variation of the functional: case of complex surfaces
+Let S be a compact complex surface. (So, we set X = S when n = 2.) We will compute the
+differential of the functional L : HS −→ [0, +∞) defined on the Hermitian cone of S. Let ω ∈ HS.
+Then, TωHS = C∞
+1, 1(S, R), so we will compute the differential
+dωL : C∞
+1, 1(S, R) −→ R
+by computing the derivative of L(ω + tγ) w.r.t. t ∈ (−ε, ε) at t = 0 for any given real (1, 1)-form γ.
+9
+
+Lemma 4.1 The differential at ω of the map HS ∋ ω �→ θ0, 1
+ω
+= Λω(¯∂ω) is given by
+(dωθ0, 1
+ω )(γ) = d
+dt|t=0Λω+tγ(¯∂ω + t ¯∂γ) = ⋆(γ ∧ ⋆¯∂ω) + Λω(¯∂γ),
+while the differential at ω of L is given by
+(dωL)(γ) = 2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂
+�
+⋆ (γ ∧ ⋆¯∂ω) + Λω(¯∂γ)
+�
+,
+for every form γ ∈ C∞
+1, 1(S, R), where ⋆ = ⋆ω is the Hodge star operator defined by the metric ω.
+Before giving the proof of this lemma, we recall the following result from [DP22] that will be
+used several times in the sequel.
+Lemma 4.2 ([DP22], Lemmas 3.5 and 3.3) For any complex manifold X of any dimension n ≥ 2,
+for any bidegree (p, q) and any C∞ family (αt)t∈(−ε, ε) of forms αt ∈ C∞
+p, q(X, C) with ε > 0 so small
+that ω + tγ > 0 for all t ∈ (−ε, ε), the following formulae hold:
+d
+dt
+����
+t=0
+(Λω+tγαt) = Λω
+�dαt
+dt
+����
+t=0
+�
+− (γ ∧ ·)⋆
+ω α0 = Λω
+�dαt
+dt
+�����
+t=0
+�
++ (−1)p+q+1 ⋆ω (γ ∧ ⋆ωα0).
+The former of the above equalities appears as such in Lemma 3.5 of [DP22], while the latter
+equality follows from the former and from formula (27) of Lemma 3.3 of [DP22] which states that
+⋆ω(η ∧ ·) = (η ∧ ·)⋆
+ω ⋆ω for any (1, 1)-form η on X.
+Indeed, in our case, taking η = γ we get
+¯η = γ since γ is real. Moreover, composing with ⋆ω on the right and using the standard equality
+⋆ω⋆ω = (−1)p+q Id on (p, q)-forms, we get ⋆ω(γ ∧ ·)⋆ω = (−1)p+q (γ ∧ ·)⋆
+ω on (p, q)-forms.
+Proof of Lemma 4.1. The formula for (dωθ0, 1
+ω )(γ) is an immediate consequence of Lemma 4.2 applied
+with αt = ¯∂ω + t ¯∂γ (hence also with (p, q) = (1, 2)). We further get:
+(dωL)(γ)
+=
+d
+dt|t=0L(ω + tγ) = d
+dt|t=0
+�
+S
+∂θ1, 0
+ω+tγ ∧ ¯∂θ0, 1
+ω+tγ
+=
+�
+S
+∂
+�
+⋆ (γ ∧ ⋆∂ω) + Λω(∂γ)
+�
+∧ ¯∂θ0, 1
+ω
++
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂
+�
+⋆ (γ ∧ ⋆¯∂ω) + Λω(¯∂γ)
+�
+.
+This is the stated formula for (dωL)(γ) since the two terms of the r.h.s. expression are mutually
+conjugated.
+□
+We will now simplify the above expression of (dωL)(γ) starting with a preliminary observation.
+Lemma 4.3 Let (X, ω) be an n-dimensional complex Hermitian manifold and let ⋆ = ⋆ω be the
+Hodge star operator defined by ω.
+(i) For every (0, 1)-form α on X, we have:
+⋆(α ∧ ω) = iΛω(α ∧ ωn−1).
+10
+
+Moreover, if n = 2, then ⋆(α ∧ ω) = iα for any (0, 1)-form α on X.
+(ii) If n = 2, then ⋆(γ ∧ α) = iΛω(γ ∧ α) for any (1, 1)-form γ and any (0, 1)-form α on X.
+In particular, ⋆¯∂ω = iθ0, 1
+ω
+for any Hermitian metric ω on a complex surface.
+(iii) In arbitrary dimension n, for any (1, 1)-form γ and any (0, 1)-form α on X, we have:
+Λω(γ ∧ α) = (Λωγ) α + i ξα⌟γ,
+where ξα is the (unique) vector field of type (1, 0) defined by the requirement
+ξα⌟ω = iα.
+Proof. (i) From the standard formula ⋆Λω = Lω⋆ (cf. e.g. [Dem97, VI, §.5.1]) we get
+Λω = ⋆Lω⋆ on even-degreed forms and Λω = − ⋆ Lω⋆ on odd-degreed forms.
+Consequently, ⋆(α ∧ ω) = ⋆Lωα = −(⋆Lω⋆) ⋆ α = Λω(⋆α) = Λω(−(1/i) α ∧ ωn−1/(n − 1)!), where
+we used the fact that ⋆⋆ = −1 on odd-degreed forms and the standard formula (10) applied to the
+(necessarily primitive) (0, 1)-form α.
+When n = 2, we get ⋆(α ∧ ω) = iΛω(α ∧ ω) = i[Λω, Lω] α = −i(1 − 2) α = iα after using the
+general formula [Lω, Λω] = (k − n) on k-forms on n-dimensional complex manifolds.
+(ii) If n = 2, the map ω ∧ · : Λ1T ⋆X −→ Λ3T ⋆X is an isomorphism at every point of X. Since
+γ ∧ α is a 3-form, there exists a unique 1-form β (necessarily of type (0, 1)) such that γ ∧ α = ω ∧ β.
+Moreover, β = Λω(γ ∧ α) because ω ∧ Λω(γ ∧ α) = [Lω, Λω](γ ∧ α) = γ ∧ α. Indeed, ω ∧ (γ ∧ α) = 0
+for bidegree reasons (here n = 2) and [Lω, Λω] = (k − n) on k-forms.
+Thus, γ ∧ α = ω ∧ Λω(γ ∧ α). So, applying (i) for the second identity below, we get:
+⋆(γ ∧ α)
+=
+⋆(ω ∧ Λω(γ ∧ α)) = iΛω(ω ∧ Λω(γ ∧ α))
+=
+i[Λω, Lω](Λω(γ ∧ α)) = iΛω(γ ∧ α).
+For the last equality, we used again the general formula [Lω, Λω] = (k − n) on k-forms (n = 2 here).
+In order to prove the formula for ⋆¯∂ω, recall that ¯∂ω = ω ∧ θ0, 1
+ω , so we get
+⋆¯∂ω = ⋆(ω ∧ θ0, 1
+ω ) = iΛω(ω ∧ θ0, 1
+ω ) = i[Λω, Lω] θ0, 1
+ω
+= −i(1 − 2) θ0, 1
+ω ,
+where we used the first part of (ii) to get the second identity.
+(iii) Since the claimed identity is pointwise and involves only zero-th order operators, we fix an
+arbitrary point x ∈ X and choose local holomorphic coordinates about x such that at x we have
+ω =
+n�
+a=1
+idza ∧ d¯za
+and
+γ =
+n�
+j=1
+γj¯j idzj ∧ d¯zj.
+Then, Λω = −i
+n�
+j=1
+∂
+∂¯zj ⌟ ∂
+∂zj ⌟· at x. If we set α =
+n�
+j=1
+αj d¯zj (at any point), we get ξα =
+n�
+j=1
+αj
+∂
+∂zj (at
+11
+
+x) and the following equalities (at x):
+Λω(γ ∧ α)
+=
+−i
+n
+�
+j=1
+∂
+∂¯zj
+⌟ ∂
+∂zj
+⌟(γ ∧ α)
+(a)
+= −i
+n
+�
+j=1
+∂
+∂¯zj
+⌟
+�� ∂
+∂zj
+⌟γ
+�
+∧ α
+�
+=
+−i
+n
+�
+j=1
+� ∂
+∂¯zj
+⌟ ∂
+∂zj
+⌟γ
+�
+∧ α + i
+n
+�
+j=1
+� ∂
+∂zj
+⌟γ
+�
+∧
+� ∂
+∂¯zj
+⌟α
+�
+(b)
+=
+�
+n
+�
+j=1
+γj¯j
+�
+α −
+n
+�
+j=1
+αjγj¯j d¯zj = (Λωγ) α + iξα⌟γ,
+where (a) follows from (∂/∂zj)⌟α = 0 for bidegree reasons and (b) follows from (∂/∂zj)⌟γ = iγj¯j d¯zj
+and from (∂/∂¯zj)⌟α = αj.
+This proves the desired equality at x, hence at any point since x was arbitrary.
+□
+We can now derive a simplified form of the first variation of the functional L.
+Theorem 4.4 Let S be a compact complex surface on which a Hermitian metric ω has been fixed.
+(i) The differential at ω ∈ HS of the functional L : HS −→ [0, +∞) evaluated at any form
+γ ∈ C∞
+1, 1(S, R) is given by any of the following three formulae:
+(dωL)(γ)
+=
+−2 Re
+�
+S
+Λω(γ) ∂θ1, 0
+ω
+∧ ¯∂θ0, 1
+ω
+− 2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂Λω(γ) ∧ θ0, 1
+ω
++ 2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂Λω(¯∂γ)
+−2 Re
+�
+S
+i∂θ1, 0
+ω
+∧ ¯∂(ξθ0, 1
+ω ⌟γ)
+(13)
+=
+−2 Re
+�
+S
+Λω(γ) |∂θ1, 0
+ω |2
+ω dVω − 2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂Λω(γ) ∧ θ0, 1
+ω
+− 2 Re i⟨⟨∂ ¯∂θ1, 0
+ω , ∂γ⟩⟩ω
+−2 Re
+�
+S
+i∂θ1, 0
+ω
+∧ ¯∂(ξθ0, 1
+ω ⌟γ)
+(14)
+=
+−2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂Λω(γ ∧ θ0, 1
+ω ) − 2 Re i⟨⟨∂ ¯∂θ1, 0
+ω , ∂γ⟩⟩ω,
+(15)
+where ⋆ = ⋆ω is the Hodge star operator defined by the metric ω and ξθ0, 1
+ω
+is the vector field of type
+(1, 0) defined by the requirement ξθ0, 1
+ω ⌟ω = iθ0, 1
+ω .
+(ii) In particular, for any given ω ∈ HS, if we choose γ = ∂θ0, 1
+ω
++ ¯∂θ1, 0
+ω , we have
+(dωL)(γ) = −2 Re
+�
+S
+i∂θ1, 0
+ω
+∧ ¯∂
+�
+ξθ0, 1
+ω ⌟γ
+�
+= −2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂Λω(γ ∧ θ0, 1
+ω ).
+Proof. (i) From (ii) and (iii) of Lemma 4.3 applied with α := iθ0, 1
+ω , we get
+⋆(γ ∧ ⋆¯∂ω) = ⋆(γ ∧ iθ0, 1
+ω ) = i Λω(γ ∧ iθ0, 1
+ω ) = −Λω(γ) θ0, 1
+ω .
+12
+
+Formula (13) follows from this and from Lemma 4.1.
+To get (14), we first notice that ¯∂θ0, 1
+ω
+= ⋆¯∂θ0, 1
+ω
+by the standard formula (10) applied to the
+(necessarily primitive) (0, 2)-form ¯∂θ0, 1
+ω . This accounts for the first term on the r.h.s. of (14). Then,
+we transform the third term in (13) as follows:
+2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂Λω(¯∂γ)
+(a)
+=
+−2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂ ⋆ Lω ⋆ (¯∂γ)
+(b)
+= 2 Re
+�
+S
+¯∂∂θ1, 0
+ω
+∧ ⋆(ω ∧ ⋆(¯∂γ))
+(c)
+=
+2 Re i
+�
+S
+¯∂∂θ1, 0
+ω
+∧ ⋆(¯∂γ)
+(d)
+= 2 Re i
+�
+S
+⟨¯∂∂θ1, 0
+ω , ∂¯γ⟩ω dVω,
+where we used the standard identity Λω = − ⋆ Lω⋆ on odd-degreed forms to get (a), Stokes to get
+(b), part (i) of Lemma 4.3 to get (c), and the definition of ⋆ to get (d). Finally, we recall that ¯γ = γ
+since γ is real.
+Finally, (15) follows from Lemma 4.1 after using the equality ⋆(γ ∧ ⋆¯∂ω) = −Λω(γ ∧ θ0, 1
+ω ) (seen
+above in the proof of (13)) and after transforming the third term in (13) as we did above in the
+proof of (14).
+(ii) The stated choice of γ means that γ is the component (dθω)1, 1 of type (1, 1) of the primitive
+2-form dθω.
+(See (i) of Lemma 2.2 for the primitivity statement.)
+Since Λω((dθω)2, 0) = 0 and
+Λω((dθω)0, 2) = 0 for bidegree reasons, we infer that
+Λω(γ) = Λω((dθω)1, 1) = Λω(dθω) = 0.
+Therefore, the first two integrals on the r.h.s. of (13) vanish.
+Meanwhile, to handle the third integral on the r.h.s. of (13), we notice that ∂¯γ = ∂ ¯∂θ1, 0
+ω
+and
+this gives the second equality below:
+2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂Λω(¯∂γ) = 2 Re i
+�
+S
+⟨¯∂∂θ1, 0
+ω , ∂¯γ⟩ω dVω = −2 Re i||¯∂∂θ1, 0
+ω ||2
+ω = 0,
+where the first equality above followed from the proof of (14).
+Thus, the r.h.s. of formula (13) for (dωL)(γ) reduces to its last integral for this choice of γ. This
+proves the first claimed equality.
+For the same reason as above, the latter term on the r.h.s. of formula (15) for (dωL)(γ) vanishes.
+This proves the second claimed equality.
+□
+As an application of (i) of Theorem 4.4, we will now see that the differential dωL vanishes on all
+the real (1, 1)-forms γ that are ω-anti-primitive (in the sense that γ is ⟨ , ⟩ω-orthogonal to all the
+ω-primitive (1, 1)-forms, a condition which is equivalent to γ being a function multiple of ω).
+Corollary 4.5 Let S be a compact complex surface on which a Hermitian metric ω has been fixed.
+For any real-valued C∞ function f on X, we have
+(dωL)(fω) = 0.
+In particular, for any real (1, 1)-form γ on S we have
+(dωL)(γ) = (dωL)(γprim),
+where γprim is the ω-primitive component of γ in its Lefschetz decomposition.
+13
+
+Proof. Applying formula (13) with γ = fω and using the obvious equalities Λω(fω) = 2f (recall
+that dimCS = 2) and ξθ0, 1
+ω ⌟(fω) = f (iθ0, 1
+ω ), we get:
+(dωL)(fω)
+=
+−4 Re
+�
+S
+f ∂θ1, 0
+ω
+∧ ¯∂θ0, 1
+ω
+− 4 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂f ∧ θ0, 1
+ω
++2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂Λω(f ¯∂ω + ¯∂f ∧ ω) − 2 Re
+�
+S
+i∂θ1, 0
+ω
+∧ (if ¯∂θ0, 1
+ω
++ i¯∂f ∧ θ0, 1
+ω )
+=
+T1 + T2 + T3 + T4,
+(16)
+where T1, T2, T3 and T4 stand for the four terms, listed in order, on the r.h.s. of the above expression
+for (dωL)(fω).
+Computing T3, we get:
+T3 = 2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂(f θ0, 1
+ω ) + 2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂
+�
+[Λω, Lω](¯∂f)
+�
+,
+where we used the equalities Λω(¯∂ω) = θ0, 1
+ω
+(see (1)) and Λω(¯∂f) = 0 (which leads to Λω(¯∂f ∧ ω) =
+[Λω, Lω](¯∂f)).
+Now, it is standard that [Λω, Lω] = (n − k) Id on k-forms on an n-dimensional
+complex manifold, so in our case we get [Λω, Lω](¯∂f) = ¯∂f since n = 2 and k = 1. We conclude
+that ¯∂([Λω, Lω](¯∂f)) = ¯∂2f = 0, hence
+T3 = 2 Re
+�
+S
+f ∂θ1, 0
+ω
+∧ ¯∂θ0, 1
+ω
++ 2 Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂f ∧ θ0, 1
+ω
+= T4,
+where the last equality follows at once from the definition of T4.
+Thus, formula (16) translates to
+(dωL)(fω)
+=
+T1 + T2 + T3 + T4
+=
+(−4 + 4) Re
+�
+S
+f ∂θ1, 0
+ω
+∧ ¯∂θ0, 1
+ω
++ (−4 + 4) Re
+�
+S
+∂θ1, 0
+ω
+∧ ¯∂f ∧ θ0, 1
+ω
+=
+0.
+This proves the first statement.
+The second statement follows at once from the first, from the linearity of the map dωL and from
+the Lefschetz decomposition γ = γprim + (1/2) Λω(γ) ω.
+□
+We hope that it will be possible in the future to prove that any Hermitian metric ω on a compact
+complex surface that is a critical point for the functional L is actually an lcK metric.
+5
+First variation of the functional: case of dimension ≥ 3
+In this section, we suppose that the complex dimension of X is n ≥ 3. The goal is to compute the
+differential of the energy functional L introduced in Definition 3.1-(ii). Let ω be a Hermitian metric
+on X and let γ be a real (1, 1)-form. The latter can bee seen as a tangent vector to HX at ω.
+14
+
+Theorem 5.1 For any Hermitian metric ω and any real (1, 1)-form γ, we have:
+(dωL)(γ)
+=
+�
+X
+i(¯∂ω)prim ∧ (¯∂ω)prim ∧ γ ∧ ωn−4
++2Re ⟨⟨(¯∂ω)prim, (¯∂γ)prim⟩⟩ω − 2Re ⟨⟨θ0, 1
+ω
+∧ γ, (¯∂ω)prim⟩⟩ω.
+(17)
+Proof. Recall (cf. the conjugate of (4)) that (n−1) θ0, 1
+ω
+= Λω(¯∂ω) for any Hermitian metric ω. Now,
+for any real t sufficiency close to 0, ω + tγ is again a Hermitian metric on X. Taking αt = ¯∂ω + t ¯∂γ
+in Lemma 4.2, we get the second equality below:
+(n − 1) d
+dt
+����
+t=0
+θ0, 1
+ω+tγ = d
+dt
+����
+t=0
+Λω+tγ(¯∂ω + t¯∂γ) = Λω(¯∂γ) − (γ ∧ ·)⋆
+ω (¯∂ω).
+(18)
+On the other hand, taking (d/dt)|t=0 in the expression for L(ω + tγ) given in (ii) of Definition
+3.1 (with ω + tγ in place of ω), we get:
+(dωL)(γ) = d
+dt
+����
+t=0
+L(ω + tγ) = d
+dt
+����
+t=0
+�
+X
+i(¯∂ω + t¯∂γ)prim ∧ (¯∂ω + t¯∂γ)prim ∧ (ω + tγ)n−3,
+(19)
+where the subscript prim indicates the (ω + tγ)-primitive part of the form to which it is attached.
+Now, consider the Lefschetz decompositions (cf. (5)) of ¯∂ω and ¯∂γ with respect to ω:
+¯∂ω
+=
+(¯∂ω)prim + θ0, 1
+ω
+∧ ω
+¯∂γ
+=
+(¯∂γ)prim + θ0, 1
+γ
+∧ ω
+and the Lefschetz decomposition of ¯∂ω + t¯∂γ with respect to ω + tγ:
+¯∂ω + t¯∂γ
+=
+(¯∂ω + t¯∂γ)prim + θ0, 1
+ω+tγ ∧ (ω + tγ).
+By the above equations we get:
+(¯∂ω + t¯∂γ)prim = (¯∂ω)prim + θ0, 1
+ω
+∧ ω + t (¯∂γ)prim + t θ0, 1
+γ
+∧ ω − θ0, 1
+ω+tγ ∧ (ω + tγ),
+(20)
+where primitivity is construed w.r.t. the metric ω + tγ in the case of the left-hand side term and
+w.r.t. the metric ω in the case of (¯∂ω)prim and (¯∂γ)prim.
+Thanks to (20), equality (19) becomes:
+(dωL)(γ)
+=
+d
+dt
+����t=0
+�
+X
+i
+�
+(¯∂ω)prim + θ0, 1
+ω
+∧ ω + t (¯∂γ)prim + t θ0, 1
+γ
+∧ ω − θ0, 1
+ω+tγ ∧ (ω + tγ)
+�
+∧
+�
+(¯∂ω)prim + θ0, 1
+ω
+∧ ω + t (¯∂γ)prim + t θ0, 1
+γ
+∧ ω − θ0, 1
+ω+tγ ∧ (ω + tγ)
+�
+∧ (ω + tγ)n−3.
+Now,
+d
+dt
+����t=0
+�
+θ0, 1
+ω+tγ ∧ (ω + tγ)
+�
+=
+θ0, 1
+ω
+∧ γ +
+� d
+dt
+����t=0
+θ0, 1
+ω+tγ
+�
+∧ ω
+=
+θ0, 1
+ω
+∧ γ +
+1
+n − 1
+�
+Λω(¯∂γ) − (γ ∧ ·)⋆
+ω(¯∂ω)
+�
+∧ ω,
+15
+
+where formula (18) was used to get the last equality. Using this, straightforward computations yield:
+(dωL)(γ) = I1 + I1 + I2,
+(21)
+where
+I2
+=
+�
+X
+i
+�
+(¯∂ω)prim + θ0, 1
+ω
+∧ ω − θ0, 1
+ω
+∧ ω
+�
+∧
+�
+(¯∂ω)prim + θ0, 1
+ω
+∧ ω − θ0, 1
+ω
+∧ ω
+�
+∧ ωn−4 ∧ γ
+=
+�
+X
+i(¯∂ω)prim ∧ (¯∂ω)prim ∧ ωn−4 ∧ γ
+(22)
+and
+I1
+=
+�
+X
+i
+�
+(¯∂γ)prim + θ0, 1
+γ
+∧ ω − θ0, 1
+ω
+∧ γ −
+1
+n − 1
+�
+Λω(¯∂γ) − (γ ∧ ·)⋆
+ω(¯∂ω)
+�
+∧ ω
+�
+∧ (∂ω)prim ∧ ωn−3
+=
+�
+X
+i(¯∂γ)prim ∧ (∂ω)prim ∧ ωn−3 −
+�
+X
+i θ0, 1
+ω
+∧ γ ∧ (∂ω)prim ∧ ωn−3,
+(23)
+where the last equality follows from (∂ω)prim ∧ ωn−2 = 0 (a consequence of the ω-primitivity of the
+3-form (∂ω)prim) which leads to the vanishing of the products of the second and the fourth terms
+(that are multiples of ω) inside the large parenthesis with (∂ω)prim ∧ωn−3 in the integral on the first
+line of (23).
+Now, due to the ω-primitivity of the 3-form (∂ω)prim, the standard formula (10) yields:
+⋆(∂ω)prim = i (���ω)prim ∧ ωn−3,
+(24)
+where ⋆ = ⋆ω is the Hodge star operator induced by ω. Thus, (22) translates to
+I1
+=
+�
+X
+(¯∂γ)prim ∧ ⋆(¯∂ω)prim −
+�
+X
+θ0, 1
+ω
+∧ γ ∧ ⋆(¯∂ω)prim
+=
+⟨⟨(¯∂γ)prim, (¯∂ω)prim⟩⟩ω − ⟨⟨θ0, 1
+ω
+∧ γ, (¯∂ω)prim⟩⟩ω.
+This last formula for I1, together with (21) and (22), proves the contention.
+□
+Recall that we are interested in the set of critical points of L. We now notice that a suitable
+choice of γ in the previous result leads to an explicit description of this set. Since equation (17) is
+valid for all real (1, 1)-forms γ, the choice γ = ω is licit, as any other choice. We get the following
+Corollary 5.2 Let X be a compact complex manifold with dimCX = n ≥ 3 and let L be the
+functional defined in 3.1-(ii). For any Hermitian metric ω on X, we have:
+(dωL)(ω) = (n − 1) ∥(¯∂ω)prim∥2
+ω = (n − 1) L(ω).
+(25)
+Proof. Taking γ = ω in equation (17), we get:
+(dωL)(ω)
+=
+�
+X
+i(¯∂ω)prim ∧ (¯∂ω)prim ∧ ω ∧ ωn−4 + 2Re ⟨⟨(¯∂ω)prim, (¯∂ω)prim⟩⟩ω
+−2Re ⟨⟨θ0, 1
+ω
+∧ ω, (¯∂ω)prim⟩⟩ω
+=
+(n − 3)i
+�
+X
+(¯∂ω)prim ∧ (¯∂ω)prim ∧ ωn−3 + 2 ∥(¯∂ω)prim∥2
+ω − 2Re ⟨⟨θ0, 1
+ω , Λω((∂ω)prim)⟩⟩ω
+=
+(n − 1)∥(¯∂ω)prim∥2
+ω,
+16
+
+where the last equality followed from (¯∂ω)prim∧ωn−3 = −i ⋆(¯∂ω)prim (see (24)) and from Λω((∂ω)prim)) =
+0 (due to any ω-primitive form lying in the kernel of Λω).
+□
+An immediate consequence of Corollary 5.2 is the following
+Proposition 5.3 Let X be a compact complex manifold with dimCX = n ≥ 3 and let ω be a
+Hermitian metric on X.
+If ω is a critical point for the functional L defined in 3.1-(ii), then ω is lcK.
+Proof. If ω is a critical point for L, then (dωL)(γ) = 0 for any real (1, 1)-form γ on X. Taking γ = ω
+and using (25), we get (¯∂ω)prim = 0. By (ii) of Lemma 2.2, this is equivalent to ω being lcK.
+□
+The converse follows trivially from what we already know. Indeed, if ω is an lcK metric, L(ω) = 0
+(by Lemma 3.2), so L achieves its minimum at ω since L ≥ 0. Any minimum is, of course, a critical
+point.
+6
+Normalised energy functionals when dimCX ≥ 3
+We start with the immediate observation that the functional introduced in (i) of Definition 3.1 in
+the case of compact complex surfaces is scaling-invariant, so it does not need normalising.
+Proposition 6.1 Let S be a compact complex surface. The functional L : HS −→ [0, +∞), L(ω) =
+�
+X ∂θ1, 0
+ω
+∧ ¯∂θ0, 1
+ω , has the property:
+L(λω) = L(ω)
+for every constant λ > 0 and every Hermitian metric ω on S.
+Proof. Recall (cf. (2)) that θ1, 0
+ω
+= Λω(∂ω) and θ0, 1
+ω
+= Λω(¯∂ω). On the other hand, for any constant
+λ > 0 and any form α of any bidegree (p, q), we have:
+Λλωα = 1
+λ Λωα,
+as can be checked right away. Therefore, θ1, 0
+λω = θ1, 0
+ω
+and θ0, 1
+λω = θ0, 1
+ω
+for every constant λ > 0. The
+contention follows.
+□
+By contrast, the functional L : HX −→ [0, +∞) introduced in (ii) of Definition 3.1 in the case
+of compact complex manifolds X with dimCX = n ≥ 3 is not scaling-invariant. Indeed, it follows at
+once from its definition that
+L(λω) = λn−1 L(ω)
+(26)
+for every constant λ > 0 and every Hermitian metric ω on X.
+This homogeneity property of L can be used to derive a short proof of the main property of L
+that was deduced in §.5 from the result of the computation of the first variation of L, namely from
+Theorem 5.1.
+17
+
+Proposition 6.2 (Proposition 5.3 revisited) Let X be a compact complex manifold with dimCX =
+n ≥ 3 and let ω be a Hermitian metric on X. The following equivalence holds:
+ω is a critical point for the functional L defined in 3.1-(ii) if and only if ω is lcK.
+Proof. Suppose ω is a critical point for L. This means that (dωL)(γ) = 0 for every real (1, 1)-form
+γ on X. Taking γ = ω, we get the first eqsuality below:
+0 = (dωL)(ω) = d
+dt
+����t=0
+L(ω + tω) = d
+dt
+����t=0
+�
+(1 + t)n−1 L(ω)
+�
+= (n − 1) L(ω).
+Thus, whenever ω is a critical point for L, L(ω) = 0. This last fact is equivalent to the metric ω
+being lcK thanks to Lemma 3.2.
+Conversely, if ω is lcK, it is a minimum point for L, hence also a critical point, because L(ω) = 0
+by Lemma 3.2.
+□
+On the other hand, recall the following by now standard
+Observation 6.3 Let ω be a Hermitian metric on a complex manifold X with dimCX = n ≥ 2. If
+ω is both lcK and balanced, ω is K¨ahler.
+Proof.
+The Lefschetz decomposition of dω spells dω = (dω)prim + ω ∧ θ, where (dω)prim is an
+ω-primitive 3-form and θ is a 1-form on X.
+We saw in Lemma 2.2 that ω is lcK if and only if (dω)prim = 0. On the other hand, the following
+equivalences hold:
+ω is balanced
+⇐⇒ dωn−1 = 0 ⇐⇒ ωn−2 ∧ dω = 0 ⇐⇒ dω is ω-primitive ⇐⇒ dω = (dω)prim.
+We infer that, if ω is both lcK and balanced, dω = 0, so ω is K¨ahler.
+□
+It is tempting to conjecture the existence of a K¨ahler metric in the more general situation where
+the lcK and balanced hypotheses are spread over possibly different metrics.
+Conjecture 6.4 Let X be a compact complex manifold with dimCX ≥ 3. If an lcK metric ω and a
+balanced metric ρ exist on X, there exists a K¨ahler metric on X.
+Together with the behaviour of L under rescaling (see (26)), this conjecture suggests a natural
+normalisation for our functional L when n ≥ 3.
+Definition 6.5 Let X be a compact complex manifold with dimCX = n ≥ 3. Fix a Hermitian
+metric ρ on X. We define the ρ-dependent functional acting on the Hermitian metrics of X:
+�Lρ : HX → [0, +∞),
+�Lρ(ω) :=
+L(ω)
+� �
+X ω ∧ ρn−1
+�n−1,
+(27)
+where L is the functional introduced in (ii) of Definition 3.1.
+18
+
+It follows from (26) that the normalised functional �Lρ is scaling-invariant:
+�Lρ(λ ω) = �Lρ(ω)
+for every constant λ > 0. Moreover, thanks to Lemma 3.2, �Lρ(ω) = 0 if and only of ω is an lcK
+metric on X.
+We now derive the formula for the first variation of the normalised functional �Lρ in terms of the
+similar expression for the unnormalised functional L that was computed in Theorem 5.1.
+Proposition 6.6 Let X be a compact complex manifold with dimCX = n ≥ 3. Fix a Hermitian
+metric ρ on X. Then, for any Hermitian metric ω and any real (1, 1)-form γ on X, we have:
+(dω �Lρ)(γ) =
+1
+� �
+X ω ∧ ρn−1
+�n−1
+�
+(dωL)(γ) − (n − 1)
+�
+X γ ∧ ρn−1
+�
+X ω ∧ ρn−1
+L(ω)
+�
+,
+(28)
+where (dωL)(γ) is given by formula (17) in Theorem 5.1.
+Proof. Straightforward computations yield:
+(dω�Lρ)(γ)
+=
+d
+dt
+�
+1
+� �
+X(ω + tγ) ∧ ρn−1
+�n−1 L(ω + tγ)
+�
+t=0
+=
+1
+� �
+X ω ∧ ρn−1
+�n−1 (dωL)(γ)
+−
+1
+� �
+X ω ∧ ρn−1
+�2(n−1) (n − 1)
+� �
+X
+ω ∧ ρn−1
+�n−2 � �
+X
+γ ∧ ρn−1
+�
+L(ω).
+This is formula (28).
+□
+A natural question is whether the critical points of any (or some) of the normalised functionals
+�Lρ are precisely the lcK metrics (if any) on X. The following result goes some way in this direction.
+Corollary 6.7 Let X be a compact complex manifold with dimCX = n ≥ 3. Fix a Hermitian metric
+ρ on X. Suppose a Hermitian metric ω is a critical point for �Lρ. Then:
+(i) for every ρ-primitive real (1, 1)-form γ, (dωL)(γ) = 0.
+(ii) if the metric ρ is Gauduchon, (dωL)(i∂ ¯∂ϕ) = 0 for any real-valued C2 function ϕ on X.
+Proof. (i) If γ is ρ-primitive, then γ ∧ ρn−1 = 0, so formula (28) reduces to
+(dω �Lρ)(γ) =
+(dωL)(γ)
+� �
+X ω ∧ ρn−1
+�n−1.
+Meanwhile, (dω �Lρ)(γ) = 0 for every real (1, 1)-form γ since ω is a critical point for �Lρ.
+The
+contention follows.
+19
+
+(ii) Choose γ := ω + i∂ ¯∂ϕ for any function ϕ as in the statement. We get:
+0
+(a)
+=
+� �
+X
+ω ∧ ρn−1
+�n−1
+(dω�Lρ)(ω + i∂ ¯∂ϕ)
+(b)= (dωL)(ω) − (n − 1) L(ω) + (dωL)(i∂ ¯∂ϕ)
+(c)
+= (dωL)(i∂ ¯∂ϕ),
+where ω being a critical point for �Lρ gave (a), formula (28) and the metric ρ being Gauduchon (the
+latter piece of information implying
+�
+X i∂ ¯∂ϕ ∧ ρn−1 = 0 thanks to the Stokes theorem) gave (b),
+while Corollary 5.2 gave (c).
+□
+As in the case of surfaces, our hope is that it will be possible in the future to prove that any
+Hermitian metric ω on a compact complex manifold of dimension ≥ 3 that is a critical point for one
+(or all) of the normalised functionals �Lρ is actually an lcK metric.
+Concluding remarks.
+(a) Let X be a compact complex manifold with dimCX = n ≥ 3. Fix a Hermitian metric ρ on
+X and consider the set Uρ of ρ-normalised Hermitian metrics ω on X such that
+�
+X
+ω ∧ ρn−1 = 1.
+By Definition 6.5, we have �Lρ(ω) = L(ω) for every ω ∈ Uρ. Moreover, since �Lρ is scaling-invariant,
+it is completely determined by its restriction to Uρ. Let
+cρ := inf
+ω∈HX
+�Lρ(ω) = inf
+ω∈Uρ
+�Lρ(ω) = inf
+ω∈Uρ L(ω) ≥ 0.
+For every ε > 0, there exists a Hermitian metric ωε ∈ Uρ such that cρ ≤ L(ωε) < cρ + ε. Since
+Uρ is a relatively compact subset of the space of positive (1, 1)-currents equipped with the weak
+topology of currents, there exists a subsequence εk ↓ 0 and a positive (see e.g. the terminology of
+[Dem97, III-1.B.]) (1, 1)-current Tρ ≥ 0 on X such that the sequence (ωεk)k converges weakly to Tρ
+as k → +∞. By construction, we have:
+�
+X
+Tρ ∧ ρn−1 = 1.
+The possible failure of the current Tρ ≥ 0 to be either a C∞ form or strictly positive (for example in
+the sense that it is bounded below by a positive multiple of a Hermitian metric on X) constitutes
+an obstruction to the existence of minimisers for the functional �Lρ. If it eventually turns out that
+the critical points of �Lρ, if any, are precisely the lcK metrics of X, if any, they will further coincide
+with the minimisers of �Lρ. In that case, the currents Tρ will provide obstructions to the existence
+of lcK metrics on X.
+(b) The same discussion as in the above (a) can be had on a compact complex surface S using
+the (already scaling-invariant) functional L introduced in (i) of Definition 3.1 if one can prove that
+its critical points coincide with the lcK metrics on S.
+20
+
+References
+[AD15] V. Apostolov, G. Dloussky — Locally Conformally Symplectic Structures on Compact Non-
+K¨ahler Complex Surfaces — Int. Math. Res. Notices, No. 9 (2016) 2717-2747.
+[DP22] S. Dinew, D. Popovici — A Variational Approach to SKT and Balanced Metrics — arXiv:2209.12813v1.
+[Dem 84] J.-P. Demailly — Sur l’identit´e de Bochner-Kodaira-Nakano en g´eom´etrie hermitienne —
+S´eminaire d’analyse P. Lelong, P. Dolbeault, H. Skoda (editors) 1983/1984, Lecture Notes in Math.,
+no. 1198, Springer Verlag (1986), 88-97.
+[Dem97] J.-P. Demailly — Complex Analytic and Algebraic Geometry — http://www-fourier.ujf-
+grenoble.fr/ demailly/books.html
+[Gau77] P. Gauduchon — Le th´eor`eme de l’excentricit´e nulle — C.R. Acad. Sc. Paris, S´erie A, t.
+285 (1977), 387-390.
+[Ist19] ˙N. Istrati — Existence Criteria for Special Locally Conformally K¨ahler Metrics — Ann. Mat.
+Pura Appl. 198 (2) (2019), 335-353.
+[OV22] L. Ornea, M. Verbitsky — Principles of Locally Conformally Kahler Geometry — arXiv:2208.07188v2.
+[Oti14] A. Otiman — Currents on Locally Conformally K¨ahler Manifolds — Journal of Geometry
+and Physics, 86 (2014), 564-570.
+[Mic83] M. L. Michelsohn — On the Existence of Special Metrics in Complex Geometry — Acta
+Math. 143 (1983) 261-295.
+[PS22] O. Perdu, M. Stanciu — Vaisman Theorem for lcK Spaces —arXiv:2109.01000v3.
+[Vai76] I. Vaisman, — On Locally Conformal Almost K¨ahler Manifolds — Israel J. Math. 24 (1976)
+338-351.
+[Voi02] C. Voisin — Hodge Theory and Complex Algebraic Geometry. I. — Cambridge Studies in
+Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2002.
+Universit´e Paul Sabatier, Institut de Math´ematiques de Toulouse
+118, route de Narbonne, 31062, Toulouse Cedex 9, France
+Email: [email protected]
+and
+[email protected]
+21
+

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+Multi-Messenger Constraint on the Hubble Constant H0
+with Tidal Disruption Events
+Thomas Hong Tsun Wong∗
+Department of Physics, University of California, San Diego, California, 92092, USA
+(Dated: January 23, 2023)
+Tidal disruption events (TDEs), apart from producing luminous electromagnetic (EM) flares,
+can generate potentially detectable gravitational wave (GW) burst signals by future space-borne
+GW detectors. In this Letter, we propose a methodology to constrain the Hubble constant H0 by
+incorporating the TDE parameters measured by EM observations (e.g., stellar mass, black hole (BH)
+mass and spin, and other orbital parameters) into the observed TDE GW waveforms. We argue
+that an accurate knowledge of the BH spin could help constrain the orbital inclination angle, hence
+alleviating the well-known distance-inclination degeneracy in GW waveform fitting. For individual
+TDEs, the precise redshift measurement of the host galaxies along with the luminosity distance DL
+constrained by EM and GW signals would give a self-contained measurement of H0 via Hubble’s
+law, completely independent of any specific cosmological models.
+I.
+INTRODUCTION
+The method of utilizing the emissions of gravitational
+wave (GW) by compact object mergers, known as the
+“standard sirens” (well-defined sources emitting at some
+known frequencies), to measure H0 was long proposed
+[1]. From Hubble’s law:
+vH = cz = H0DL ,
+(1)
+the detected GW waveform provides constraints on DL
+while the bright electromagnetic (EM) counterpart mea-
+sures the redshift, known as the “bright siren” (an EM-
+observable “standard siren”). It was only until recently
+has this technique been implemented in the binary neu-
+tron star merger event GW170817 [2]. The uncertainty in
+H0 measurement is, however, dominated by the degener-
+acy between DL and the inclination angle η, defined here
+as the angle between the orbital angular momentum and
+the line of sight [3], of the binary system from the GW
+template-waveform fitting [2, 4], as seen for a small-angle
+approximation:
+hGW ∝ cos η
+DL
+,
+(2)
+where hGW is the detected GW strain amplitude.
+By
+incorporating the multi-messenger information of the
+event, the viewing-angle-dependent features of various
+EM emission models (e.g., gamma-ray burst and kilo-
+nova) are exploited to arbitrate the distance-inclination
+degeneracy, providing a tighter constraint on DL, and
+subsequently H0 ([4] and references therein).
+One would naturally question whether compact object
+mergers remain the sole astrophysical sources to mea-
+sure H0 in a multi-messenger approach. As long as mas-
+sive objects revolve around each other, GW emissions are
+guaranteed, therefore tidal disruption of stars by massive
+∗ Email: [email protected]
+black holes (BHs) would present themselves as viable can-
+didates due to the fact that immense EM radiation is
+released during the transient event [5–8]. When a star
+approaches the galactic central supermassive black hole
+(SMBH) at a sufficiently close distance, the tidal radius
+rT ≈ (MBH/m⋆)1/3 r⋆, where MBH, m⋆, r⋆ are the BH
+mass, stellar mass, and stellar radius, respectively, the
+star is then torn apart given that the tidal field of the
+hole exceeds the star’s self-gravity [9].
+The disrupted
+stellar material would be stretched into a debris stream,
+approximately half of it will gradually dissipate orbital
+energy into EM radiation, and eventually circularize into
+an accretion disk. Optical, near-UV, X-ray all-sky sur-
+veys have detected up to a hundred or so events, and
+one to two orders of magnitude more are expected in the
+coming decade [10, 11].
+Tidal disruption events (TDEs) could only generate
+GW bursts as the star is often disrupted within an or-
+bital timescale, i.e. the intact star does not survive an
+entire orbit to produce a full period of GW waveform [5].
+An open comprehensive living catalog of TDE GW wave-
+forms has been built to explore a wide range of parame-
+ters [8]. Given their relatively long orbital timescale prior
+to disruption (∼ 102−4 s), the characteristic GW burst
+frequency is approximately in the range of 0.1 − 10 mHz,
+which corresponds to the designed sensitivities of the up-
+coming space-borne GW detectors [12–15]. But in fact,
+most TDE GW signals are incapable of generating a large
+enough signal-to-noise ratio to trigger a detection for
+LISA [12] but would lie well within the detection limit of
+post-LISA detectors [7, 14–16]. The TDE GW observed
+rate by LISA is predicted to remain half a dozen or so
+for the entire four-year mission [7] as the characteristic
+strains of the events are weak given typical TDE param-
+eters, which are estimated as [5, 8]:
+hGW ∼ 10−22
+�
+DL
+20 Mpc
+�−1
+×
+β
+� r⋆
+R⊙
+�−1 � m⋆
+M⊙
+�4/3 � MBH
+106 M⊙
+�2/3
+,
+(3)
+arXiv:2301.08407v1  [astro-ph.HE]  20 Jan 2023
+
+2
+where rp is the pericenter radius and β = rp/rT is the
+penetration parameter [17] (quantifying how deeply the
+orbit penetrates into the BH gravitational potential well).
+The pessimistic observed rate by LISA inspires us to
+explore the possibility of using a very limited number of
+events to independently measure H0.
+We hereby pro-
+pose using TDE EM observations to constrain as many
+parameters as possible prior to GW waveform fitting, re-
+sulting in a remarkably improved GW constraint on the
+luminosity distance DL. This work focuses on gathering
+the cumulative modeling effort in obtaining TDE param-
+eters, exploring their intercorrelations, and most impor-
+tantly, proposing the first methodology to constrain H0
+via TDEs. Prior to detecting GW signals, using simu-
+lated waveforms to run the following analysis in an at-
+tempt to constrain H0 would inevitably lead to a circu-
+lar argument, therefore this letter serves as a primal in-
+vestigation on the foundational idea of exploiting multi-
+messenger signals of TDEs to measure H0.
+In Section II, we illustrate the recent progress in con-
+straining all waveform-dependent TDE parameters by
+EM observations with physical modeling, laying the foun-
+dational work to further propose the novel approach to
+alleviate the distance-inclination degeneracy using the
+constrained BH spin parameters, the methodology is
+then presented with an estimation on which parameter(s)
+would most dominate the H0 uncertainty. In Section III,
+we discuss possible ways to further improve the precision
+of H0 determination.
+II.
+MULTI-MESSENGER CONSTRAINTS ON DL
+Given the ability to localize TDEs with the current
+multi-wavelength surveys, the redshifts of the host galax-
+ies can be comfortably measured with high certainty
+[10, 11]. In order to estimate an accurate Hubble con-
+stant H0, the problem lies in constraining the luminosity
+distance DL from their GW counterparts.
+Unfortunately, in order to accurately match the ob-
+served waveforms with the templates, one could not
+rely solely on the approximate peak amplitude in Eq.(3)
+which depends mainly on three parameters.
+Provided
+the number of waveform-dependent parameters, there is
+only hope to constrain DL, and hence H0, if most, if
+not all, these TDE parameters could be constrained to
+some extent with EM observations. We hereby show the
+parameter inter-dependencies (summarized in Fig.2) and
+how they are expected to yield a constrained H0.
+A.
+EM Constraints on TDE Parameters
+1.
+Three main parameters: MBH, m⋆, β
+TDE-host black hole mass is one of the easiest to infer
+since there are a few MBH-galaxy relations available [18–
+20] and simulations/models specifically for TDE-specific
+scenarios [5, 21–25]. The common TDE EM-GW detec-
+tion rate is highly limited [7], a more accurate constraint
+on the MBH of the few detectable events could perhaps be
+computed in a case-by-case TDE-model-dependent man-
+ner instead of using the global galaxy relations, even if
+the latter has a slightly better constraint than the former.
+Focusing on the disruption of main-sequence (MS)
+stars, the mass-radius relation is given by [26], such that
+all r⋆-dependence turns into m⋆ accordingly.
+MCMC
+method that fits both the peak luminosity and the
+color temperature of the observed TDEs could con-
+strain m⋆ down to a ∼few % uncertainty (but some-
+times a lot higher) [23, 25].
+It is indeed a challeng-
+ing task to check whether the mass constraint is ac-
+curate given there is yet to exist another independent
+EM-measurement, additional GW waveform information
+could complement the deficiency based on the approxi-
+mate duration τ/frequency f of the burst [5, 8]:
+f ∼ 1
+τ ≈ 10−4 Hz × β3/2
+� m⋆
+M⊙
+�1/2 � r⋆
+R⊙
+�−3/2
+.
+(4)
+Orbital parameters remain the most challenging-to-
+constrain variables as the TDE observables do not de-
+pend as sensitively as they do on the masses. The β de-
+pendency on TDE light curves is investigated with hydro-
+dynamical simulations [21], resulting in co-dependency
+on both masses. A relatively weaker constraint on β is
+through analyzing the probability distribution among the
+EM-observed TDE population [27]. Both works provided
+corresponding analytical formulae. For the high signal-
+to-noise TDE detections by LISA, a lower bound on β
+(e.g., βmin ≳ 10 for MBH = 106 M⊙) can be imposed [7].
+Given the rough EM-dependence, it could potentially be
+slightly more beneficial to further constrain β from the
+TDE GW waveform as the overall shape of the polar-
+izations and the duration of the GW burst change when
+β is varied [8]. The EM constraint can simply be used
+as the prior knowledge with a large uncertainty between
+βmin ≳ β ≥ βmax, where βmax happens at rp = rSch, i.e.,
+all stellar materials are swallowed at pericenter passage
+thus EM signal detection is unlikely.
+It is important to note that these three parameters
+are not completely independent, e.g., a less massive, i.e.,
+more compact, star could not be tidally disrupted by a
+very massive BH, but will instead be swallowed whole,
+i.e. rT ≤ rSch, where rSch is the Schwarzschild radius of
+the hole. Some regions in the parameter space are thus
+automatically ruled out for any EM-observable event.
+2.
+Degeneracy-breaking parameters:
+Black hole spin and inclination angle: aBH, θ, η
+Upon first glance, the spin properties of the host BH
+might seem like a sub-dominant factor, but the way they
+correlate with the orbital inclination angle η could ul-
+timately lead to a probable alleviation of the known
+distance-inclination degeneracy.
+
+3
+𝜃
+x
+y
+z (los)
+x
+y
+z (los)
+x
+y
+z (los)
+incoming orbital plane
+BH spin
+𝜂
+accretion 
+disk plane
+multiple 
+windings
+observed
+prediction
+simulation
+FIG. 1. Schematic illustration of how a typical η orbit
+would evolve into an accretion disk of specific ori-
+entation given a BH spin offset. When the (large) BH
+spin orientation sufficiently differs from the orbital angular
+momentum of the stellar orbit, the disrupted stream debris
+would evolve away from the initial orbital plane, resulting in a
+stream collision somewhere else (red explosion) [21]. The po-
+sition of the intersection could constrain the final orientation
+of the circularized accretion disk, where then the viewing-
+angle dependent model may be applied [28]. The z-axis is set
+to be the line of sight (los).
+The dimensionless spin magnitude and the spin orien-
+tation are defined by 0 ≤ aBH ≡ cJ/GM 2
+BH ≤ 1 and the
+angle between the spin vector and the stellar orbital an-
+gular momentum, 0◦ (prograde) ≤ θ ≤ 180◦ (retrograde),
+respectively. Spin parameters could be constrained with
+the observed light curve at peak accretion [29] (most sen-
+sitive for high β) or with X-ray reverberation technique
+when the accretion disk is formed [30]. If an observed
+TDE has a rapidly spinning host BH, the launching of a
+relativistic jet via the Blandford-Znajek mechanism [31]
+would be advantageous in further constraining the BH
+spin parameters [32].
+The parameter η has never been investigated with EM
+observables as the incoming orbit of the star has (nearly)
+no influence on the multiwavelength/multi-epoch signa-
+tures as we could not see lights at the exact disruption
+phase.
+However, when GW is included as part of the
+multi-messenger analysis, η is of utmost importance.
+The first-ever analysis on incorporating viewing-angle-
+dependent models was used for the case of binary neu-
+tron star merger GW170817 [4], where physical models
+of gamma-ray burst and kilonova are used to constrain
+the system inclination by modeling their light curves and
+spectra-photometry. We propose that a similar approach
+could be implemented with the work of [28, 33], in which
+the TDE spectral features depend mainly on the view-
+ing angle [34]. Assuming that the X-ray and optical/UV
+emissions originate from the inner part of the accretion
+disk and the disk luminosity reprocessed by the expand-
+ing outflow, respectively [35], when viewing the TDE sys-
+tem edge-on, the intrinsic X-ray emission from the disk
+will be reprocessed by the optically thick outflow, opti-
+cal/UV luminosity would dominate the observed spec-
+trum; when viewing the TDE system face-on, we would
+then expect to look into the optically thin funnel and
+see a stronger X-ray luminosity from the exposed in-
+ner disk. Given that both optical and X-ray luminosi-
+ties are measured for many TDE candidates, their ratios
+Loptical/LX−ray could potentially indicate the approxi-
+mate inclination angle of the geometrically thick outer
+accretion disk [28, 33], i.e., increase in viewing angle of
+the disk generally augments the optical-to-X-ray lumi-
+nosity ratio.
+To link the orientations of the disk and the stellar or-
+bital plane, for BH spins that are aligned with the orbital
+plane’s normal (θ = 0), we could safely assume that the
+stellar materials would on average remain on its incoming
+orbital plane due to symmetry, such that the luminosity
+ratio could directly be used to infer η. For cases where
+the direction of the moderate/high BH spin is signifi-
+cantly offset from the orbital plane’s normal (incoming
+stellar orbits are often randomly oriented), relativistic
+precession for close encounters (high β) would induce de-
+flections of the debris stream out of the initial orbital
+plane, leading to a certain period when the TDE flare
+is not observable [36]. When the stream eventually self-
+intersects and dissipates its orbital energy during circu-
+larization, it is unlikely that the circularized disk will lie
+on the original orbital plane [37]. Hence, in order to uti-
+lize this analysis, both BH spin parameters should not
+be neglected, their relations are illustrated by the stream
+evolution in Fig.1 and indicated in Fig.2. When the spin
+parameters are constrained by the modeling discussed in
+Section BH spin, the simulation [36] could then be im-
+plemented to predict how the initial orbit plane would
+undergo multiple windings and end up on the final ac-
+cretion disk plane where the stream intersection finally
+happens, hence yielding the orbital inclination angle η
+through forward modeling.
+
+4
+The spin parameters have to be constrained by EM
+observations as they are the input parameters for de-
+termining the orbital inclination angle η, meaning that if
+aBH and θ are fitted through GW waveform, the distance-
+inclination degeneracy would still remain.
+3.
+Other orbital parameters: e, φ
+From the GW waveform simulation [8], the strain am-
+plitude dependence of the orbital eccentricity e is approx-
+imately an order of magnitude smaller than β and θ. Al-
+beit the mild sensitivity in e, implying the insignificant
+contribution to the uncertainty of DL, EM constraints
+are still possible. The common assumption is that most
+TDE stars have a roughly parabolic (e ≈ 1) flyby orbit
+[38].
+Hyperbolic orbits (e ≳ 1) are automatically not
+taken into consideration since the stellar materials are
+unbound after the disruption and would not produce a
+detectable EM signal; and near-circular orbits are highly
+unlikely based on loss cone dynamics [39]. The eccen-
+tricity is then essentially constrained to some values in
+between given by the fallback rate [40].
+There exists one orbital orientation parameter φ, which
+is the angle between the stellar pericenter axis and the
+projection of the line of sight onto the orbital plane [8],
+being the least dependent of all. This angle can hardly
+be constrained by EM observations nor TDE models and
+shall not possess any prior in the waveform fitting. (All
+angles discussed θ, η, and φ are defined identically to [8].)
+B.
+Luminosity Distance DL and H0 Estimate
+1.
+Key Methodology
+Using suitable TDE models to fit the correspond-
+ing observables discussed in Section II A, all seven EM-
+constrained parameters (MBH, m⋆, β, aBH, θ, η, e) would
+have their corresponding probability density functions
+(PDFs) obtained through simulations and model-fitting.
+As for DL and φ, they are expected to freely vary during
+the waveform fitting, and no prior assumption should be
+made on DL to prevent any bias. With the knowledge
+to constrain the inclination angle η as an input parame-
+ter, the distance-inclination degeneracy is expected to be
+relieved to a very large extent.
+All nine variables and their corresponding uncertain-
+ties would be used to generate a huge catalog of GW
+waveforms [8], centering at the constrained parameter
+values to avoid exploring the large parameter space. Note
+that some parameter values are prohibited (rT ≤ rSch),
+e.g., large MBH for disruption of small m⋆, high β for
+specific MBH and m⋆. The theoretically generated wave-
+forms would then be fitted with that of the observed in
+a similar manner as in [41], yielding a PDF of DL. The
+PDF of DL would translate directly to the PDF of H0 us-
+ing Eq.1. Setting this PDF as the likelihood in Bayesian
+𝑴𝐁𝐇
+𝒎⋆
+𝜷
+𝑒
+𝑎$%
+𝜃
+𝜂
+𝜙
+𝐷&
+ℎ'(,*+,-.(𝑡)
+EM constraints
+ℎ'(,+/0(𝑡)
+fitting
+𝐷&
+𝑧+/0
+𝐻1
+To be fitted
+𝐻1
+𝐻1
+prior
+posterior
+likelihood
+EM 
+constraint
+FIG. 2. Graphical model describing how the relations
+amongst TDE parameters and how EM and GW ob-
+servations are combined to yield H0 from a single
+TDE. The main TDE parameters (MBH, m⋆, β) are indi-
+cated by blue circles. Dashed arrows illustrate how param-
+eters are obtained via other parameters, which involve some
+simulations/models and additional EM observables (e.g., light
+curves and spectra). EM-constrained parameters combining
+with the GW waveform fitting process would give the PDF
+of DL, then with the observed host galaxy redshift, the PDF
+of H0 (likelihood) is found. The H0 likelihood and cosmo-
+logically determined prior then give rise to the posterior (the
+final determination of H0 by a single TDE). The filled boxes
+indicate the outputs of each procedure.
+formalism [2] and the H0 from other cosmological studies
+[42, 43] as the prior, the posterior H0 is expected to peak
+near the prior value while eliminating the other H0 peaks
+derived from DL. The graphical description is shown in
+Fig.2.
+2.
+Bottleneck in H0 Measurement Uncertainty
+As long as the parameters are entangled in such a
+complicated manner (Fig.2), what we could do is, from
+an order-of-magnitude point of view, estimate which pa-
+rameter(s) would predominantly contribute to the uncer-
+tainty σDL.
+Fig.3 shows the main TDE parameters that impact the
+GW burst amplitudes (the exact waveform of hGW here is
+less important than those of LIGO events as TDEs often
+
+5
+−20
+−10
+0
+10
+20
+t − tburst [103 s]
+10−24
+10−23
+10−22
+10−21
+10−20
+hGW
+MBH = 105 M⊙, m⋆ = 1 M⊙, β = 1
+MBH = 106 M⊙, m⋆ = 1 M⊙, β = 1
+MBH = 106 M⊙, m⋆ = 1 M⊙, β = 2
+MBH = 106 M⊙, m⋆ = 1 M⊙, β = 5
+MBH = 107 M⊙, m⋆ = 1 M⊙, β = 1
+MBH = 107 M⊙, m⋆ = 10 M⊙, β = 1
+FIG. 3.
+TDE parameters that dominate the depen-
+dency on GW waveform amplitude |hGW|.
+Amongst
+the seven EM-constrained parameters, varying these three pa-
+rameters: MBH (solid), m⋆ (dotted), and β (dashed), would
+fluctuate hGW to a large extent. All waveforms are centered
+at tburst, the time when the peak amplitude is reached. The
+other parameters are chosen as follows: aBH = 0, θ = 0, e = 1,
+η = 0, DL = 20 Mpc. Plotted from simulation results [8].
+only generate single bursts), while the rest either become
+important only in extreme scenarios (high β or near-
+maximal BH spin) or are always subdominant.
+These
+three parameters, coincidentally, are often the input pa-
+rameters in most simulations (as seen from the number of
+arrows pointing out of them in Fig.2), thus their uncer-
+tainties are cumulative and are projected onto the rest.
+Amongst them, we believe the penetration parameter β
+ought to contribute the largest uncertainty of H0. Even
+though MBH is used thrice to determine other parameters
+(while twice by β), MBH is typically better constrained
+than β [22, 23, 27], unless for very massive BHs where
+the detectable β range can be as narrow as order of unity.
+σH0 is found to be dominated by the distance-inclination
+degeneracy [2, 4], implying that ση should dominate over
+the rest. Given that β is used to determine η, σβ should
+in turn dominate.
+It is understandable as the magni-
+tude of the off-plane precession sensitively depends on
+how close the stellar debris orbits around the spinning
+BH [36]. σMBH would then be the next dominating un-
+certainty.
+The few hundred TDE GW waveforms in the presently
+enlarging library [8] have a resolution too low in the 9-
+dimensional parameter space to yield a reasonable fitting.
+This should immediately raise the question: What is the
+approximate number of waveforms required to result in
+a reasonable fit, which then translates into a reasonable
+H0 precision? If a uniform search in parameter space is
+implemented, the number of waveforms generated would
+skyrocket as the number of parameters increase. Hav-
+ing established that each parameter affects the waveform
+to different extents, it is only sensible to vary densely
+on the parameters of dominant contributions, such as
+MBH, β, and η. Adaptive resolution on which parame-
+ters to explore should precede uniformly increasing the
+total number of waveforms across all parameter spaces in
+the catalog. Ultimately, the goal of the multi-messenger
+analysis is to better constrain DL, not finding the best-fit
+TDE parameters.
+III.
+DISCUSSION
+As predicted by [5, 7, 8, 16], even the optimistic TDE
+GW detection rate by LISA is expected to remain a few
+for the entire duration of the mission. It is therefore of
+utmost importance that the analyses of the few limited
+multi-messenger TDE observations could be maximized,
+stressing the power of this methodology to independently
+measure H0 with a handful of events. To strengthen the
+constraining power of TDE parameters as a whole, the
+GW burst signal during disruption could trigger the im-
+mediate follow-up EM observations such that light from
+the pre-peak epoch can be captured.
+When DECIGO
+and the other next-generation spaceborne GW detectors
+are eventually in operation, the expected thousands to
+millions of TDE detections might in turn place EM ob-
+servation as the bottleneck of the multimessenger era,
+but by then a statistically significant measurement of H0
+from TDEs should already be obtained.
+If the uncer-
+tainty on DL, hence H0, could be reduced even by some
+small portion, after incorporating EM constraints with
+this proposed methodology, this would then conclusively
+demonstrate the functionality of TDE multi-messenger
+H0 measurement, while placing the development of TDE
+modeling at the bottleneck of the analysis.
+For typical cases, σβ would be dominant, still, there
+are certain possible ways to further constrain β:
+by
+brute force, we would benefit from a GW TDE triggering
+of pre-peak high-cadence EM observation [21]; more β-
+sensitive observable could be found with improved mod-
+eling; or exploiting the potentially huge detectable TDE
+population by post-LISA interferometers, then the β-
+distributions [27] could directly constrain H0 and not the
+individual β in each event.
+Given the modeling complication and intertwining re-
+lations among parameters, measuring H0 with TDEs is
+clearly a non-trivial task and would likely require a col-
+laborative effort in the field. All in all, it is manifest that
+the proliferating EM and GW detections of TDEs and
+more comprehensive TDE simulations in the next decade
+should lead to both precise and accurate measurements
+of the Hubble constant in addition to the standard siren
+approach.
+
+6
+ACKNOWLEDGMENTS
+I thank S.K. Li, Paul C.W. Lai, Lars L. Thomsen, and
+George M. Fuller for useful comments and discussions.
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+Effective and Efficient Training for Sequential Recommendation
+Using Cumulative Cross-Entropy Loss
+Fangyu Li,1 Shenbao Yu, 2 Feng Zeng, 3 Fang Yang 1*
+1 2 3 Department of Automation, Xiamen University, Xiamen China
+{lifangyu, yushenbao}@stu.xmu.edu.cn, {zengfeng, yang}@xmu.edu.cn
+Abstract
+Increasing research interests focus on sequential recom-
+mender systems, aiming to model dynamic sequence repre-
+sentation precisely. However, the most commonly used loss
+function in state-of-the-art sequential recommendation mod-
+els has essential limitations. To name a few, Bayesian Per-
+sonalized Ranking (BPR) loss suffers the vanishing gradi-
+ent problem from numerous negative sampling and prediction
+biases; Binary Cross-Entropy (BCE) loss subjects to nega-
+tive sampling numbers, thereby it is likely to ignore valuable
+negative examples and reduce the training efficiency; Cross-
+Entropy (CE) loss only focuses on the last timestamp of the
+training sequence, which causes low utilization of sequence
+information and results in inferior user sequence representa-
+tion. To avoid these limitations, in this paper, we propose to
+calculate Cumulative Cross-Entropy (CCE) loss over the se-
+quence. CCE is simple and direct, which enjoys the virtues of
+painless deployment, no negative sampling, and effective and
+efficient training. We conduct extensive experiments on five
+benchmark datasets to demonstrate the effectiveness and effi-
+ciency of CCE. The results show that employing CCE loss on
+three state-of-the-art models GRU4Rec, SASRec, and S3-Rec
+can reach 125.63%, 69.90%, and 33.24% average improve-
+ment of full ranking NDCG@5, respectively. Using CCE, the
+performance curve of the models on the test data increases
+rapidly with the wall clock time, and is superior to that of
+other loss functions in almost the whole process of model
+training.
+Introduction
+With the rapid development of recurrent neural networks
+(RNN), transformer, graph neural network (GNN), convo-
+lutional neural network (CNN), and other deep neural net-
+works, sequential recommendation models based on user in-
+teraction records are becoming increasingly popular in rec-
+ommender systems. For instance, GRU4Rec (Hidasi et al.
+2016), GRU4Rec+ (Hidasi and Karatzoglou 2018), and
+NARM (Li et al. 2017) are based on RNN; SASRec (Kang
+and McAuley 2018), BERT4Rec (Sun et al. 2019), S3-Rec
+(Zhou et al. 2020), and NOVA-BERT (Liu et al. 2021) are
+based on transformer; SR-GNN (Wu et al. 2019) and Caser
+(Tang and Wang 2018) are based on GNN and CNN, respec-
+tively.
+*Corresponding author. Email-address: [email protected]
+In order to unleash the full potential of the sequence rec-
+ommendation model, it needs to match a suitable loss func-
+tion that plays an essential role in determining the effective-
+ness and efficiency of model training. However, existing loss
+functions used in sequential recommendation have their own
+defects. For example, one of the popular methods, GRU4Rec
+utilizes BPR (Rendle et al. 2009) or TOP1 loss as the ob-
+jective function, which suffers from the gradient vanishing
+problem (Hidasi and Karatzoglou 2018).
+We focus on two rarely discussed issues about loss func-
+tions. First, most loss functions only calculate the loss on the
+last timestamp of the training sequence, which ignores the
+natural sequential properties of sequence data. Fig. 1 gives
+an illustrative example, where Fig. 1(a) and Fig. 1(b) show
+the difference in loss calculation of GRU4Rec and SASRec,
+the former involves only the last timestamp while the latter
+covers all timestamps. Fig. 1(c) visualizes the NDCG@10
+scores of GRU4Rec on each timestamp of the user sequence
+(the length is fixed to 50) of Yelp data, using three dif-
+ferent loss functions. As shown in Fig. 1(c), the vanilla
+GRU4Rec optimizes the loss on the last timestamp of train-
+ing sequence, so it achieves its highest performance at the
+last timestamp (the 48th), but has the poorest performance
+at other timestamps including the validation (the 49th) and
+test data (the 50th). Instead, the GRU4Rec model trained
+with BCE loss optimizes all timestamps of the training se-
+quence, which results in performance improvements over
+vanilla GRU4Rec on the validation and test data. This obser-
+vation indicates that only calculating the last timestamp loss
+in the objective function cannot guarantee the accuracy of
+the intermediate timestamp, which causes low utilization of
+sequence information and generates inferior user sequence
+representation.
+Second, negative sampling is a widely-used approach to
+improve performance for sequential recommendation. Cor-
+respondingly, the loss functions, e.g. BCE used in SASRec,
+considers a small number of negative examples for each
+timestamp in each user sequence, which indicates that it in-
+volves tiny parts of the negative samples and is likely to
+ignore some informative negative examples. On the other
+hand, increasing the number of negative samples will re-
+duce the computational efficiency, hence the trade-off be-
+tween the model effectiveness and efficiency is hard to bal-
+ance when employing negative sampling in model training.
+arXiv:2301.00979v1  [cs.IR]  3 Jan 2023
+
+Transfor
+mer
+Transfor
+mer
+Transfor
+mer
+Embedding Layer
+Prediction Layer
+S1
+S2
+S3
+S4
+Out1
+Out2
+Out3
+R1
+R2
+R3
+Calculate Loss on the All Timestamps
+GRU
+Embedding Layer
+Prediction 
+Layer
+S1
+S2
+S3
+S4
+Out1
+Out2
+Out3
+R3
+Calculate Loss on the Last Timestamp
+GRU
+GRU
+(b) SASRec model architecture.
+(a) GRU4Rec model architecture.
+(c) Simplified experimental results of GRU4Rec with different losses.
+Figure 1: The architectures of GRU4Rec & SASRec and performance comparison of three loss functions. We displays the
+average NDCG@10 scores of GRU4Rec using three loss functions, at each timestamp on the Yelp dataset. The sequence length
+is fixed to 50, with the 49th and 50th timestamps represent the validation and test item respectively.
+To tackle these problems, in this paper, we propose a
+novel Cumulative Cross-Entropy (CCE) loss that jointly
+considers all timestamps in the training process and all neg-
+ative samples for loss function calculation without negative
+sampling (see also the performance of the proposed method
+in Fig. 1(c)). In addition, CCE sufficiently covers the gra-
+dient of the item embedding matrix by each item’s softmax
+score. Furthermore, the proposed model employs the mask-
+ing strategy for the varied length of user sequence to guar-
+antee the training efficiency.
+We validate our method on three typical sequential recom-
+mendation models (i.e., GRU4Rec, SASRec, and S3-Rec)
+on five benchmark datasets from different domains. Exper-
+imental results show that our method obtain average im-
+provements of 125.63%, 69.90%, and 33.24% in terms of
+full ranking NDCG@5 for GRU4Rec, SASRec, and S3-Rec,
+respectively. Specifically, GRU4Rec trained with CCE loss
+can markedly improve the NDCG@5 score by 266.67% over
+vanilla GRU4Rec on the Toys dataset (McAuley et al. 2015).
+The main contributions are threefold. First, we identify
+limitations in the existing loss function used by sequen-
+tial recommendation models. Second, we designed Cumu-
+lative Cross-Entropy loss, which extends the cross-entropy
+to all timestamps of the training sequence and can effec-
+tively solve the limitation of timestamp and negative sam-
+pling. Lastly, we conduct extensive experiments on five real-
+world datasets, demonstrating significant improvements in
+HIT@k and NDCG@k metrics over existing state-of-the-art
+methods.
+Related Work
+According to the sequence timestamps involved in the loss
+computation, we divide the loss functions used in existing
+sequential recommendation models into three categories. To
+the best of our knowledge, this issue has not received much
+attention in existing studies.
+Last Timestamp Loss Family
+It refers to the loss function that only involves the last
+timestamp of the training sequence. Generally, most neural
+network-based sequential recommendation models belong
+to this family. The first sequential recommendation method
+based on RNN is GRU4Rec (Hidasi et al. 2016), which uti-
+lizes the Gated Recurrent Units (GRU) and employs several
+pointwise and pairwise ranking losses - such as BPR, TOP1,
+and CE, which only calculate the loss of the last timestamp.
+Besides, the improved GRU4Rec+ (Hidasi and Karatzoglou
+2018) argues that the original pairwise loss function used
+in GRU4Rec likely causes the gradient vanishing problem,
+thereby proposes the improved listwise loss function BPR-
+max and TOP-max. Most recent works that are influenced by
+GRU4Rec directly adopt or adapt BPR loss, e.g., the hierar-
+chical gating networks HGN (Ma, Kang, and Liu 2019), the
+GNN-based model MA-GNN (Ma et al. 2020) and STEN
+(Li et al. 2021). Besides, some models use CE loss as the
+objective function, such as NARM (Li et al. 2017), STAMP
+(Liu et al. 2018), SMART SENSE (Jeon et al. 2022), and SR-
+GNN (Wu et al. 2019). To alleviate the item cold-start prob-
+lem, Mecos (Zheng et al. 2021) uses CE loss to optimize a
+meta-learning task. Besides, a recent work (Petrov and Mac-
+donald 2022) utilizes the LambdaRank (Burges 2010) loss
+function, which still belongs to the last timestamp family.
+Masked Language Model Loss Family
+The Masked Language Model (MLM) (Devlin et al. 2018)
+loss is derived from the cloze task (Taylor 1953), and the ob-
+jective is to accurately predict the item that randomly mask
+in input sequence. Recent work adopted the idea of MLM
+and employs MLM loss in sequential recommendation. For
+example, BERT4Rec (Sun et al. 2019), utilizes BERT (De-
+vlin et al. 2018) to model user behavior; NOVA-BERT (Liu
+et al. 2021) introduces an attention mechanism that suf-
+ficiently leverages side information to guide and preserve
+item representations invariant in its vector space. However,
+the item masking methods sacrifice much training time to
+achieve good performances.
+
+All Timestamp Loss Family
+As the name suggests, it considers all timestamps of the
+training sequence in loss computation. To the best of our
+knowledge, the BCE loss is the mainly member in this fam-
+ily besides CCE loss proposed in this paper. It is employed
+in the CNN-based model Caser (Tang and Wang 2018), the
+attention-based model SASRec (Kang and McAuley 2018),
+RKSA (Ji et al. 2020), ELECRec (Chen, Li, and Xiong
+2022) and CAFE (Li et al. 2022), and the state-of-the-art
+self-supervised learning model S3-Rec (Zhou et al. 2020).
+Note that S3-Rec uses BCE loss at its fine-tuning stage, and
+utilizes item attributes and Mutual Information Maximiza-
+tion (MIM) to capture fusion between context data and se-
+quence data at the pre-training stage. In addition, the genera-
+tor module in ELECRec extends the CE loss to ALL Times-
+tamp, but its role in the loss calculation does not ignore the
+mask item as the BCE loss does. There is a paucity of discus-
+sions on the training objective of BCE loss. In our opinion,
+all timestamp loss is able to take full advantage of the prop-
+erties of sequence data, that is, the input under the current
+timestamp is the label of the previous timestamp. However,
+the BCE loss is inevitably affected by negative sampling,
+and the number of negative samples will affect its perfor-
+mance and computational efficiency.
+Typical models and Loss Functions in
+sequential recommendation
+We first formulate the problem of sequential recommenda-
+tion, then introduce two most representative model struc-
+tures of neural network-based sequential recommendation
+models and the most commonly used loss functions, i.e.
+BPR, TOP1, BCE, and CE.
+Problem Statement
+Suppose that there are a set of users U =
+�
+u1, u2, ..., u|U|
+�
+and a set of items I =
+�
+i1, i2, ..., i|I|
+�
+, where |U| and
+|I| denote the the number of users and items, respectively.
+In the sequential recommendation, we mainly focus on the
+user’s historical interaction records. Therefore, we formulate
+a user sequence S1:n = (S1, S2, ..., Sn) based on interaction
+records in chronological order, where n denotes the length
+of user sequence and St denotes the user interaction item at
+timestamp t. We first define two kinds of sequential recom-
+mendation models below:
+Rn = flast(S1:n),
+(1)
+R1:n = fall(S1:n),
+(2)
+where flast and fall are models that adopt the last times-
+tamp loss and all timestamp loss, respectively. Rn
+=
+�
+rn,1, rn,2, ..., rn,|I|
+�
+denotes the outputs of all items at
+timestamp n, where rn,t is the prediction score of item it
+at timestamp n. R1:n = (R1, R2, ..., Rn) is the result on all
+timestamps.
+Next, we define the embedding layer and prediction layer,
+which are the typical operations in the sequential recommen-
+dation. Given a sequence input with the fixed-length l, the
+input sequence of the embedding layer (i.e., S1:l) is trans-
+formed to the embedding vector E1:l = (e1, e2, ..., el) ∈
+Rl×e by the embedding matrix We ∈ R|I|×e. In addition,
+the prediction layer is an unbiased dense layer with a weight
+matrix W T
+e , which shares the weight matrix with the embed-
+ding layer. We now proceed to inntroduce the GRU4Rec and
+SASRec models, as well as the corresponding loss fuctions.
+GRU4Rec
+Model Architecture
+GRU4Rec is one of the most classi-
+cal sequential recommendation models, which utilizes GRU
+to model the user sequence and output a sequence represen-
+tation. Given three components of the GRU, i.e., the update
+gate z, the candidate hidden state ˆh and the reset gate r, the
+hidden state ht ∈ Rd can be calculated as:
+ht = zt ˆht + (1 − zt)ht−1.
+(3)
+In Eq. 3, we have:
+zt = σ(Wzet + Uzht−1),
+(4)
+ˆht = σ(Whet + Uh(rt ⊙ ht−1)),
+(5)
+rt = σ(Wret + Urht−1),
+(6)
+where Wz,r,h ∈ Rd×e and Uz,r,h ∈ Rd×d are the weight
+matrices, respectively. The last hidden state hl of the GRU
+is the vector that represents the input sequence S1:l, which
+passes through the prediction layer to get the final result
+Rl = hlW T
+e =
+�
+rl,1, rl,2, ..., rl,|I|
+�
+.
+Loss Function
+There are three loss function of vanilla
+GRU4Rec, i.e., BPR loss (Rendle et al. 2009), TOP1 loss,
+and CE loss. Here we give the calculation method of BPR
+and TOP1 as follows:
+Lbpr = − 1
+Ns
+Ns
+�
+neg=1
+log σ(rl,pos − rl,neg),
+(7)
+Ltop1 = 1
+Ns
+Ns
+�
+neg=1
+σ(rl,neg − rl,pos),
+(8)
+where Ns is the number of negative samples. rl,pos, rl,neg
+are the scores of the positive item and negative item at the
+last timestamp l, respectively. Note that we omit the regular-
+ization term for readability since it has nothing to do with the
+following discussion. To simplify the formula, we use b+,−
+to represent the prediction bias of (rl,pos − rl,neg) and b−,+
+to denote (rl,neg − rl,pos). We then examine their gradients
+w.r.t. the score of positive item rl,pos as follows:
+∂Lbpr
+∂rl,pos
+= − 1
+Ns
+Ns
+�
+neg=1
+(1 − σ(b+,−)) ,
+(9)
+∂Ltop1
+∂rl,pos
+= 1
+Ns
+Ns
+�
+neg=1
+σ(b−,+) (1 − σ(b−,+)) .
+(10)
+Obviously, the vanishing gradient problem will occur for
+both loss functions when the number of negative samples Ns
+increases. In addition, the prediction bias b+,− for BPR (or
+b1,+ for TOP1) that tends to infinity also induces the vanish-
+ing gradient problem. In practice, due to the huge size of the
+
+negative set, the case of prediction bias occurs frequently.
+Therefore, GRU4Rec+ proposed the improved BPR-max
+and TOP1-max losses via applying softmax scores on nega-
+tive examples, which can be calculated as follows:
+Lbpr−max = − log
+Ns
+�
+neg=1
+snegσ(b+,−),
+(11)
+Ltop1−max =
+Ns
+�
+neg=1
+snegσ(b−,+),
+(12)
+where sneg is the softmax score of the negative examples
+ineg. We also examine their gradients w.r.t. the score of pos-
+itive item rl,pos:
+∂Lbpr−max
+∂rl,pos
+= −
+�Ns
+neg=1 snegσ(b+,−) (1 − σ(b+,−))
+�Ns
+neg=1 snegσ(b+,−)
+,
+(13)
+∂Ltop1−max
+∂rl,pos
+= −
+Ns
+�
+neg=1
+snegσ(b−,+)(1 − (σ(b−,+)) .
+(14)
+Through the softmax score sneg, the new loss can miti-
+gate the vanishing gradient problem. However, as we men-
+tioned, the trade-off between the model effectiveness and
+efficiency is hard to balance when employing negative sam-
+pling in model training. Meanwhile, the sampling operation
+may skip informative negative samples.
+SASRec
+Model Architecture
+The SASRec model is the first to in-
+troduce Transformer (Vaswani et al. 2017) into sequential
+recommendation. SASRec stacks two layers of transformer
+encoders. For readability, we only introduce the single layers
+of the transformer encoder block. Before the transformer en-
+coder, SASRec adds a position vector P1:l ∈ Rl×e, thereby
+the final input is ˆE = E1:l + P1:l. Then it use Multi-Head
+Self-Attention (MH) layer to learn the asymmetric interac-
+tions and make the model more flexible, which consists of
+multiple independent self-attention layers (SA) and trans-
+form by a weight matrix WO ∈ Re×e:
+MH( ˆE) = [SA1( ˆE), SA2( ˆE), · · · , SAH( ˆE)]WO, (15)
+SAj( ˆE) = attention( ˆEWQj, ˆEWKj, ˆEWV j),
+(16)
+attention(Q, K, V ) = softmax
+�QKT
+√e
+�
+V,
+(17)
+where WQj, WKj, WV j ∈ Re×e/H are the linear projection
+matrix that scales the input ˆE into a small space. Note that in
+the case of self-attention, the queries Q, keys K, and values
+V all equal to the input ˆE. To satisfy the nature of sequence
+data, SASRec cut off the connection of Qi and Kj(j > i)
+in the attention calculation (Eq. 17). The multi-head self-
+attention layer aggregate all previous item embedding with
+adaptive weights and is still a linear model. Therefore, to
+endow the model with nonlinear, SASRec applies a point-
+wise two-layer feed-forward network F with ReLU (Nair
+and Hinton 2010) activation function:
+F( ˆE) = ReLU(MH( ˆE)W1)W2.
+(18)
+To avoid overfitting, dropout (Srivastava et al. 2014) and
+layer normalization (Ba, Kiros, and Hinton 2016) are used
+for the input of both modules (MH and F). Further, to sta-
+bilize training, a residual connection (He et al. 2016) is ap-
+plied.
+g(x) = x + Dropout(g(LayerNormalization(x))),
+(19)
+where g(x) is the multi-head self-attention layer or point-
+wise feed-forward network. Finally, through the prediction
+layer, the result of SASRec is R1:l = F( ˆE)W T
+e .
+Loss Function
+SASRec adopts the binary cross-entropy
+(BCE) loss as the objective function, and here we use mask
+to simplify the objective function.
+Lbce = −
+l
+�
+t=1
+MASK [log σ(rt,pos) + log σ(1 − rt,neg)] ,
+(20)
+where MASK = (mask1, mask2, ..., maskl) is the mask
+vector, maskt is False when St in the sequence S1:l is the
+mask item, and True otherwise. We can find that the main
+difference in loss function between GRU4Rec and SASRec
+is the cumulative term of time. Intuitively, this loss allows
+more positive samples to participate in the optimization pro-
+cess. However, it depends on the negative sampling oper-
+ation, and randomly generates only one negative item for
+each timestamp. Further, we give the gradient w.r.t the score
+of positive item rt,pos and negative item rt,neg as follows:
+∂Lbce
+∂rt,pos
+= −maskt(1 − σ(rt,pos)),
+(21)
+∂Lbce
+∂rt,neg
+= maskt(1 − σ(1 − rt,neg)).
+(22)
+As we can see, the gradient coincides with the objective of
+the sequential recommendation. However, the majority of
+negative items do not participate in the loss calculation due
+to the sampling strategy, which means that they contribute
+little to the update of model parameters. Therefore, BCE
+is essentially prone to lose information. Intuitively, adding
+more negative examples can alleviate this problem, but it
+would spend much more time on sampling operation.
+Our Method : Cumulative Cross-Entropy Loss
+Based on the above discussions, we observe that, instead
+of average loss, adaptive loss via softmax function may be
+more suitable for sequential recommendation. In this sense,
+the Cross-Entropy (CE) loss is a natural choice. Its calcula-
+
+tion and gradient can be described as follows:
+Lce = − log
+exp (rl,pos)
+�|I|
+j=1 exp (rl,j)
+,
+(23)
+∂Lce
+∂rl,pos
+=
+exp (rl,pos)
+�|I|
+j=1 exp (rl,j)
+− 1,
+(24)
+∂Lce
+∂rl,j
+=
+exp (rl,j)
+�|I|
+j=1 exp (rl,j)
+.
+(25)
+Note that without sampling, CE loss aggregates the predic-
+tion score of the whole item size, which contains the whole
+negative example set. Compared with BCE, the CE loss is
+more suitable for sequential recommendation for the follow-
+ing reasons: 1) Sequential recommendation can be regarded
+as a multi-classification task, and the softmax function used
+in CE loss was born for this; 2) The gradient of CE loss
+can cover the whole item set in a single step. 3) CE avoids
+negative sampling, and hence refrains from difficulties aris-
+ing therefrom, such as the additional time cost of sampling.
+Therefore, CE can improve the training efficiency and re-
+duces the risk of information loss.
+However, the current form of CE loss used in sequential
+recommendation only focuses on the last timestamp. In this
+paper, we directly extend it to all timestamps, and propose
+a novel Cumulative Cross-Entropy loss, which is calculated
+as follows:
+Lcce = −
+l
+�
+t=1
+MASK log
+exp (rt,pos)
+�|I|
+j=1 exp (rt,j)
+.
+(26)
+The idea of CCE is simple and direct. It revises the short-
+sighted training objective of CE, and takes the advantage of
+BCE that perform loss calculation on each timestamp of the
+sequence; Further, it avoids the negative sampling operation
+in BCE, and calculates gradient on the entire item set like
+CE. Extensive experiments verify the effectiveness of the
+CCE loss.
+Experiments
+We conduct extensive experiments on five benchmark
+datasets to validate the effectiveness and efficiency of the
+proposed CCE loss, aiming to answer the following research
+questions. RQ1: How does the CCE loss perform when em-
+ployed in the state-of-the-art models? RQ2: How efficient is
+the training of the models using the CCE loss? RQ3: How
+does the CCE loss perform across all timestamps?
+Experiments Setup
+Datasets
+We use five public benchmark datasets collected
+from three real-world platforms, namely, three sub-category
+datasets on Amazon1 (McAuley et al. 2015): Beauty, Sports
+and Toys; a business recommendation dataset Yelp2; and a
+music artist recommendation dataset LastFM3 (Cantador,
+Brusilovsky, and Kuflik 2011). Note that we only use the
+transaction records after January 1st, 2019 in Yelp.
+1http://jmcauley.ucsd.edu/data/amazon/links.html
+2https://www.yelp.com/dataset
+3https://grouplens.org/datasets/hetrec-2011/
+Table 1: Statistics of five datasets after preprocessing
+Dataset
+Sports
+Toys
+Yelp
+Beauty
+LastFM
+# of sequences
+35598
+19412
+30431
+22362
+1090
+# of items
+18357
+11924
+20033
+12101
+3646
+# of iteractions
+296337
+167597
+316454
+198502
+52551
+Average length
+16.14
+14.06
+15.80
+16.40
+14.41
+Density
+0.05%
+0.07%
+0.05%
+0.07%
+1.32%
+Data Processing
+Following the recent and state-of-the-
+arts in sequential recommendation (Kang and McAuley
+2018; Zhou et al. 2020; Tang and Wang 2018; Sun et al.
+2019), we divide a given dataset into train, validation, and
+test sets according to the leave-one-out strategy. In addi-
+tion, to reproduce the pre-training model S3-Rec, we pre-
+processed the original datasets as follows. (1) We remove
+users and items with less than five interaction records. (2)
+We group the interaction records by users and sort them
+chronologically. (3) We keep the user sequence with the
+fixed-length l. After preprocessing, the statistics of the five
+datasets are summarized in Table 1.
+Baseline Methods
+Since most sequential recommenda-
+tion models only output results at the final timestamp Rl, we
+here choose three representative models, which are not only
+able to output all timestamp results R1:l, but also equipped
+with stable and superior performance:
+• GRU4Rec (Hidasi et al. 2016). which is the first to apply
+GRU to model user interaction sequence for the session-
+based recommendation.
+• SASRec (Kang and McAuley 2018). which is a
+transformer-based model, using a multi-head attention
+mechanism to learn the asymmetric interactions and
+make the model more flexible.
+• S3-Rec (Zhou et al. 2020). which is the first to introduce
+self-supervised learning to the sequential recommenda-
+tion.
+For comparative loss functions, we choose CE as the rep-
+resentative of the last timestamp loss function since it per-
+forms better than BPR loss and Top1 loss in our preliminary
+experiments. Besides, we use BCE as the representative of
+all timestamp loss function. Note that we ignore the masked
+language model loss due to its large training cost.
+Implementation Details
+To reproduce the sequential rec-
+ommendation models GRU4Rec, SASRec, and S3-Rec, we
+use the open-source of S3-Rec code4 and RecBole5 repo.
+The hyperparameters of these models are set as suggested
+in the original paper. For each dataset, the fixed length of
+the input sequence is set to 50, the size of the item em-
+beddings is 64. Besides, we use the Adam optimizer with
+the default learning rate of 0.001, parameters β1 and β2 are
+set to 0.9 and 0.999, respectively. We train models for 150
+epochs with the early stop strategy6. We save the optimal
+model based on the evaluation metrics on the validation set
+4https://github.com/RUCAIBox/CIKM2020-S3Rec
+5https://github.com/RUCAIBox/RecBole
+6We terminate the training when the evaluation metric does not
+improve for ten consecutive epochs.
+
+Table 2: Comparing three loss functions with respect to the performance of GRU4Rec, SASRec, and S3-Rec on five datasets. Best
+results are in boldface, and the best one between Lbce and Lce is indicated by underline. “Improve” denotes the improvement
+over the best performance of Lbce (or Lce), while the degradation cases are marked with ↓.
+Dataset
+Metric
+GRU4Rec
+SASRec
+S3-Rec
+Lbce
+Lce
+Lcce
+Improve.
+Lbce
+Lce
+Lcce
+Improve.
+Lbce
+Lce
+Lcce
+Improve.
+Sports
+HR@5
+0.0100
+0.0099
+0.0221
+121.00%
+0.0216
+0.0168
+0.0380
+75.93%
+0.0217
+0.0325
+0.0456
+40.31%
+HR@10
+0.0184
+0.0163
+0.0357
+94.02%
+0.0330
+0.0229
+0.0541
+63.94%
+0.0359
+0.0478
+0.0642
+34.31%
+HR@20
+0.0297
+0.0253
+0.0548
+84.51%
+0.0491
+0.0330
+0.0752
+53.16%
+0.0567
+0.0709
+0.0908
+28.07%
+NDCG@5
+0.0063
+0.0064
+0.0143
+123.44%
+0.0147
+0.0117
+0.0267
+81.63%
+0.0137
+0.0213
+0.0311
+46.01%
+NDCG@10
+0.0090
+0.0085
+0.0187
+107.78%
+0.0184
+0.0137
+0.0318
+72.83%
+0.0182
+0.0262
+0.0371
+41.60%
+NDCG@20
+0.0118
+0.0107
+0.0235
+99.15%
+0.0225
+0.0162
+0.0371
+64.89%
+0.0234
+0.0320
+0.0438
+36.88%
+Toys
+HR@5
+0.0128
+0.0097
+0.0420
+228.13%
+0.0430
+0.0385
+0.0736
+71.16%
+0.0409
+0.0568
+0.0791
+39.26%
+HR@10
+0.0236
+0.0153
+0.0597
+152.97%
+0.0613
+0.0485
+0.0989
+61.34%
+0.0641
+0.0796
+0.1096
+37.69%
+HR@20
+0.0401
+0.0229
+0.0834
+107.98%
+0.0862
+0.0616
+0.1299
+50.70%
+0.0998
+0.1119
+0.1492
+33.33%
+NDCG@5
+0.0081
+0.0065
+0.0297
+266.67%
+0.0288
+0.0291
+0.0533
+83.16%
+0.0261
+0.0398
+0.0566
+42.21%
+NDCG@10
+0.0116
+0.0083
+0.0354
+205.17%
+0.0347
+0.0323
+0.0615
+77.23%
+0.0335
+0.0472
+0.0664
+40.68%
+NDCG@20
+0.0157
+0.0102
+0.0414
+163.69%
+0.0410
+0.0356
+0.0693
+69.02%
+0.0425
+0.0553
+0.0764
+38.16%
+Yelp
+HR@5
+0.0128
+0.0094
+0.0211
+64.84%
+0.0166
+0.0101
+0.0232
+39.76%
+0.0206
+0.0178
+0.0290
+40.78%
+HR@10
+0.0220
+0.0164
+0.0367
+66.82%
+0.0273
+0.0174
+0.0379
+38.83%
+0.0354
+0.0311
+0.0474
+33.90%
+HR@20
+0.0378
+0.0273
+0.0603
+59.52%
+0.0499
+0.0275
+0.0623
+24.85%
+0.0552
+0.0498
+0.0756
+36.96%
+NDCG@5
+0.0080
+0.0055
+0.0134
+67.50%
+0.0106
+0.0064
+0.0151
+42.45%
+0.0126
+0.0115
+0.0184
+46.03%
+NDCG@10
+0.0109
+0.0078
+0.0184
+68.81%
+0.0140
+0.0087
+0.0198
+41.43%
+0.0173
+0.0157
+0.0243
+40.46%
+NDCG@20
+0.0149
+0.0105
+0.0244
+63.76%
+0.0184
+0.0112
+0.0259
+40.76%
+0.0223
+0.0204
+0.0314
+40.81%
+Beauty
+HR@5
+0.0161
+0.0223
+0.0489
+119.28%
+0.0358
+0.0401
+0.0694
+73.07%
+0.0379
+0.0577
+0.0753
+30.50%
+HR@10
+0.0266
+0.0343
+0.0695
+102.62%
+0.0573
+0.0537
+0.0932
+62.65%
+0.0614
+0.0830
+0.1031
+24.22%
+HR@20
+0.0447
+0.0514
+0.0998
+94.16%
+0.0878
+0.0719
+0.1286
+46.47%
+0.0979
+0.1203
+0.1440
+47.09%
+NDCG@5
+0.0100
+0.0147
+0.0342
+132.65%
+0.0235
+0.0291
+0.0492
+69.07%
+0.0232
+0.0389
+0.0529
+35.99%
+NDCG@10
+0.0133
+0.0185
+0.0408
+120.54%
+0.0305
+0.0355
+0.0568
+60.00%
+0.0307
+0.0471
+0.0619
+31.42%
+NDCG@20
+0.0179
+0.0228
+0.0484
+112.28%
+0.0381
+0.0381
+0.0657
+72.44%
+0.0400
+0.0565
+0.0721
+27.61%
+LastFM
+HR@5
+0.0248
+0.0211
+0.0339
+36.69%
+0.0266
+0.0083
+0.0450
+69.17%
+0.0431
+0.0339
+0.0422
+-2.09% ↓
+HR@10
+0.0468
+0.0312
+0.0459
+-1.92% ↓
+0.0404
+0.0156
+0.0587
+45.30%
+0.0688
+0.0541
+0.0789
+14.68%
+HR@20
+0.0624
+0.0495
+0.0606
+-2.88% ↓
+0.0550
+0.0284
+0.0862
+56.73%
+0.1220
+0.0881
+0.1349
+10.57%
+NDCG@5
+0.0161
+0.0138
+0.0222
+37.89%
+0.0179
+0.0049
+0.0310
+73.18%
+0.0273
+0.0197
+0.0262
+-4.03% ↓
+NDCG@10
+0.0232
+0.0171
+0.0262
+12.93%
+0.0223
+0.0073
+0.0354
+58.74%
+0.0356
+0.0260
+0.0381
+7.02%
+NDCG@20
+0.0272
+0.0218
+0.0299
+9.93%
+0.0259
+0.0105
+0.0423
+63.32%
+0.0491
+0.0346
+0.0519
+5.70%
+at the training stage, then report their performances on the
+test set. Note that for the pre-training model S3-Rec, we use
+the reproduced model offered by its source code, and retrain
+it at the fine-tuning stage. All experiments are conducted us-
+ing 10-cores of an Intel i9-10900K CPU, 24GB of memory
+and an NVIDIA GeForce RTX 3090 GPU.
+Evaluation Metrics
+To evaluate the performance of se-
+quential recommendation models, we adopt the top-k Hit
+Ratio (HIT@k, k=5, 10, 20) and top-k Normalized Dis-
+counted Cumulative Gain (NDCG@k, k=5, 10, 20), which
+are commonly used in previous studies (Hidasi et al. 2016;
+Kang and McAuley 2018; Zhou et al. 2020). The details of
+the metrics can be found in (Krichene and Rendle 2020).
+Recent work on sampling strategies (Dallmann, Zoller, and
+Hotho 2021; Krichene and Rendle 2020) found that under
+the same sampling test set, the results of the evaluation met-
+rics are inconsistent when using different sampling strategy.
+To avoid inconsistency, we report the full ranking metrics.
+Experimental Results
+Overall Results (RQ1)
+Table 2 shows the performance of
+the GRU4Rec, SASRec, and S3-Rec using CCE, BCE, and
+CE, respectively. We observe that the CCE loss improves
+the best performance of BCE (or CE) for all models in most
+cases. In addition, we perform t-test on the results, which
+shows that the performance of all models using the proposed
+CCE loss are significantly different from that of using BCE
+or CE (at significant level p < .001). Note that there are only
+4 out of 90 cases, where results produced by the CCE loss
+has very slight performance decrease (up to 4.03%).
+For GRU4Rec, compared with BCE and CE, the pro-
+posed CCE loss greatly promotes the performance of the
+model. The average improvements on five datasets in
+terms of HR@5, HR@10, HR@20, NDCG@5, NDCG@10,
+NDCG@20 are 113.99%, 82.90%, 68.66% 125.63%,
+103.05% and 89.76% respectively. Interestingly, the CCE
+loss brings an astonishing 266.67% improvement at
+NDCG@5 on Toys. In addition, experiments show that the
+GRU4Rec with CCE can achieve better performance on
+Sports, Yelp, Beauty, and LastFM than the original SASRec,
+which indicates that the loss function has great influence on
+model performance.
+For SASRec, our CCE loss achieves an overall increase
+in terms of all metrics on five datasets. Specifically, the av-
+erage improvements in terms of HR@5, HR@10, HR@20,
+NDCG@5, NDCG@10, NDCG@20 are 65.82%, 54.41%,
+46.38%, 69.90%, 62.05%, and 62.09%, respectively. Com-
+pared with the GRU4Rec and SASRec models, although S3-
+Rec with BCE (or CE) obtains the best performance, the
+CCE loss still shows a substantial improvement for S3-Rec
+in terms of the six metrics, i.e., the average improvements
+are 29.75% (HR@5), 28.96% (HR@10), 25.73% (HR@20),
+33.24% (NDCG@5), 32.24% (NDCG@10), and 29.83%
+(NDCG@20), respectively.
+
+(a) Sports
+(b) Toys
+(c) Yelp
+GRU4Rec
+SASRec
+S3-Rec
+(d) Beauty
+(e) LastFM
+Figure 2: The performance curve (NDCG@10) of GRU4Rec, SASRec and S3Rec using different loss functions on the test data
+during training process.
+(a) Sports
+(b) Toys
+(c) Yelp
+GRU4Rec
+(d) Beauty
+(e) LastFM
+Figure 3: The performance of GRU4Rec using different loss functions at all timestamps on the five datasets.
+Training Efficiency (RQ2)
+We evaluate the training effi-
+ciency of our approach from two aspects, as suggested in
+(Kang and McAuley 2018). Fig. 2 displays the NDCG@10
+scores on the test sets during the training process of baseline
+models with different loss functions on the five benchmark
+datasets. We also show the training speed, which counts the
+average time consumption for one training epoch (second-
+s/epoch) (see the bottom-right corner of each graph). As can
+be seen from Fig. 2, compared with the BCE and CE loss,
+despite sharing the similar training speeds for all models, the
+performance curve of the models with CCE on the test data
+increases rapidly as the wall clock time increases, as well as
+dominating the models with other loss functions for nearly
+the entire training process. For example, the SASRec+CCE
+takes about 100 seconds to reach a much higher value of
+NDCG@10 (i.e., 0.035) on Sports, while spends 12.24 sec-
+onds for one epoch, which is close to BCE (11.21s/epoch)
+and CE (12.62s/epoch). In summary, we argue that the CCE
+loss can effectively and efficiently help model training.
+Performance on All Timestamps (RQ3)
+In this section,
+we extend the results in Fig. 1c to five benchmark datasets.
+As shown in Fig. 3, the vertical axes represent NDCG@10
+scores of GRU4Rec, while the horizontal axes represent the
+whole timestamp of the input sequence. The last two times-
+tamps refer to the validation and test item, respectively,
+where the model performance drops drastically in nature.
+On Beauty, Sports, Toys, and Yelp, CCE has a very signifi-
+cant boost across all timestamps, which shows that CCE can
+better guarantee the accuracy of the intermediate process of
+model inference. For the LastFM dataset, CCE has only a
+slight improvement over BCE in the training sequence. This
+result may explain why it does not show great advantages on
+the test data. Intuitively, a loss function that is able to guar-
+antee the accuracy for all timestamps of training sequence
+can effectively improve the recommendation accuracy.
+Conclusion
+In this paper, we address the issue of loss function design
+in sequential recommendation models. We point out that the
+whole training sequence should be considered when calcu-
+lating the loss, rather than the last timestamp. Meanwhile,
+avoiding negative sampling can improve the training effi-
+ciency and accuracy of recommendations. We propose a
+novel cumulative cross-entropy loss and apply it to three
+typical models, i.e., GRU4Rec, SASRec, and S3Rec. Exper-
+iments on five benchmark datasets demonstrate its effective-
+ness. We hope that this work can inspire the design of loss
+function in the subsequent research on sequence recommen-
+
+dation models and contribute to effective and efficient train-
+ing for sequential recommendation.
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+

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+arXiv:2301.13784v1  [math.RT]  31 Jan 2023
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+NATE HARMAN AND ANDREW SNOWDEN
+Abstract. Galois categories can be viewed as the combinatorial analog of Tannakian cat-
+egories. We introduce the notion of pre-Galois category, which can be viewed as the combi-
+natorial analog of pre-Tannakian categories. Given an oligomorphic group G, the category
+S(G) of finitary smooth G-sets is pre-Galois. Our main theorem (approximately) says that
+these examples are exhaustive; this result is, in a sense, a reformulation of Fra¨ıss´e’s the-
+orem. We also introduce a more general class of B-categories, and give some examples of
+B-categories that are not pre-Galois using permutation classes. This work is motivated by
+certain applications to pre-Tannakian categories.
+Contents
+1.
+Introduction
+1
+2.
+Oligomorphic groups
+4
+3.
+Combinatorial tensor categories
+5
+4.
+Pre-Galois categories
+11
+5.
+Categories of atoms
+14
+6.
+Fra¨ıss´e theory
+18
+7.
+Examples from relational structures
+22
+References
+25
+1. Introduction
+1.1. Background. The famous Tannakian reconstruction theorem says that an algebraic
+group can be recovered from its representation category. To be a bit more precise, fix an
+algebraically closed field k. A pre-Tannakian category is a k-linear abelian category equipped
+with a symmetric tensor structure satisfying some axioms. A Tannakian category is a pre-
+Tannakian category C equipped with a fiber functor ω, i.e., a faithful exact tensor functor
+to finite dimensional vector spaces. The motivating example of a Tannakian category is the
+category Repk(G) of finite dimensional representations of an algebraic group G/k; the fiber
+functor is simply the forgetful functor. The main theorem of Tannakian categories states
+these examples are essentially exhaustive: if C is a Tannakian category then C is equivalent
+to Repk(G), where G is the (pro-algebraic) automorphism group of ω. See [DM] for details.
+It is not so easy to construct pre-Tannakian categories that are not (super-)Tannakian,
+but a number of interesting examples are known, including Deligne’s interpolation categories
+[Del], the Delannoy category [HSS], and the Verlinde category [Ost]. A major problem in
+the field of tensor categories is to better understand the pre-Tannakian landscape.
+Date: January 31, 2023.
+1
+
+2
+NATE HARMAN AND ANDREW SNOWDEN
+There is a combinatorial analog of Tannakian reconstruction, in the form of Grothendieck’s
+Galois theory. A Galois category is a category C equipped with a functor ω to finite sets
+satisfying certain axioms (see Definition 4.11). The motivating example of a Galois category
+is the category of finite G-sets, for a group G. The main theorem of Galois categories states
+that these examples are essentially exhaustive: if C is a Galois category then C is equivalent
+to the category of smooth (=discrete) G-sets, where G is the (profinite) automorphism group
+of ω. Grothendieck applied this theorem to construct the ´etale fundamental group.
+Conspicuously absent from the combinatorial side is an analog of pre-Tannakian categories.
+The purpose of this paper is to fill this gap: we define this class of categories, prove one
+main theorem about them, and construct some interesting examples.
+1.2. Pre-Galois categories. The following is our combinatorial analog of pre-Tannakian
+categories:
+Definition 1.1. A category B is pre-Galois if the following conditions hold:
+(a) B has finite co-products (and thus an initial object 0).
+(b) Every object of B is isomorphic to a finite co-product of atoms, i.e., objects that do
+not decompose under co-product.
+(c) If X is an atom and Y and Z are other objects, then any map X → Y ∐ Z factors
+uniquely through Y or Z.
+(d) B has fiber products and a final object 1.
+(e) Any monomorphism of atoms is an isomorphism.
+(f) If X → Z and Y → Z are maps of atoms then X ×Z Y is non-empty (i.e., not 0).
+(g) The final object 1 is atomic.
+(h) Equivalence relations in B are effective (see Definition 4.8).
+□
+The above axioms are motivated by properties of the category of finite G-sets, for a group
+G. “Atoms” should be thought of as transitive G-sets. The first three axioms basically say
+that objects admit a finite “orbit decomposition” which behaves in the expected manner.
+We define a B-category to be one satisfying axioms (a)–(e). This turns out to be a very
+nice class of categories already. For example, we show that every B-category is balanced
+(Corollary 3.13) and has finite Hom sets (Proposition 3.17). Axiom (e) is somewhat subtle,
+but these nice properties of B-categories depend on it.
+Of the remaining three axioms, (f) is clearly the most important: in a sense, it is easy to
+explain all failures of (g) and (h), but this is not the case for (f). We say that a B-category is
+non-degenerate if it satisfies (f) and (g). Non-degeneracy implies a number of nice properties,
+such as existence of co-equalizers. Axiom (h) ensures that quotients are well-behaved.
+One can match properties of pre-Galois categories and pre-Tannakian categories, to some
+extent. Axiom (a) corresponds to additivity on the pre-Tannakian side. Both pre-Galois
+and pre-Tannakian categories are finitely complete and co-complete. Axiom (h) corresponds
+to the first isomorphism theorem on the pre-Tannakian side.
+Axiom (b) corresponds to
+the finite length condition on the pre-Tannakian side. The co-product and product in a
+pre-Galois category correspond to the direct sum and tensor product in a pre-Tannakian
+category. Axiom (g) corresponds to the pre-Tannakian axiom that the unit object is simple.
+Finally, (f) corresponds to the fact that in a pre-Tannkain category the tensor product of
+non-zero objects is non-zero.
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+3
+1.3. Examples. For any group G, the category S(G) of finite G-sets is a pre-Galois category,
+and this is the motivating example. One might try to construct other examples by considering
+(possibly infinite) G-sets with finitely many orbits. This does not work in general since a
+product of two such G-sets need not have finitely many orbits. For instance, G acting on
+itself by left multiplication has one orbit, but the orbits of G on G × G are in bijection with
+G itself.
+It turns out that the above idea can be made to work in at least one situation, however.
+Recall that an oligomorphic group is a permutation group (G, Ω) such that G has finitely
+many orbits on Ωn for all n ≥ 0. The simplest example of an oligomorphic group is the infinite
+symmetric group. Model theory, and the theory of Fra¨ıss´e limits in particular, provides many
+more examples. See [Cam1] for general background. Given an oligomorphic group G, we
+define S(G) to be the category of sets equipped with an action of G that is smooth (every
+stabilizer is open in the natural topology) and which has finitely many orbits.
+It turns
+out that this category is closed under products; this is a consequence of the oligomorphic
+condition. It is not hard to show that S(G) is in fact pre-Galois.
+The above examples admit a mild generalization: we define a class of topological groups
+called admissible groups, which include profinite groups and oligomorphic groups, and we
+associate a pre-Galois category S(G) to such G. From the topological perspective, the key
+finiteness property of G is Roelcke pre-compactness. We review this theory in §2; a more
+detailed treatment can be found in [HS1, §2].
+In Example 7.3 we give a non-trivial example of a degenerate B-category using a non-
+Fra¨ıss´e class of relational structures. It would be interesting if one could give a more direct
+construction of such an example.
+1.4. The main theorem. The following is our main result on pre-Galois categories.
+Theorem 1.2 (Theorem 6.15). Let B be a category. The following are equivalent:
+(a) B is pre-Galois and has countably many isomorphism classes.
+(b) B is equivalent to S(G) for some first-countable admissible group G.
+The countability hypotheses above should not be necessary, but we impose them to make
+the proof and exposition easier. Theorem 6.15 in fact does a bit more than the above theorem,
+in that it accommodates all (countable) non-degenerate B-categories; in other words, we still
+obtain a classification result when we do not impose Definition 1.1(h). The non-degeneracy
+condition seems essential, however.
+1.5. Overview of proof. Let B be a B-category, and let A be the full subcategory of Bop
+spanned by atoms. We show that B can be recovered from A, and exactly characterize the
+categories A that arise in this manner (we call them A-categories). The key point in the
+proof of Theorem 1.2 is that A is a Fra¨ıss´e category, meaning it is the kind of category to
+which the categorical version of Fra¨ıss´e’s theorem applies. This theorem produces a universal
+homogeneous ind-object Ω in A. We show that G = Aut(Ω) is naturally an admissible group,
+and that B is equivalent to S(G).
+The correspondence between A- and B-categories is also useful for producing examples
+of B-categories: indeed, it is easy to construct A-categories by taking classes of relational
+structures, and one can then convert them to B-categories. We follow this plan in §7.
+
+4
+NATE HARMAN AND ANDREW SNOWDEN
+1.6. Motivation. As stated above, a major problem in tensor category theory is under-
+standing pre-Tannakian categories. In a recent paper [HS1], we made a bit of progress on
+this problem: we constructed a pre-Tannakian category Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Repk(G, µ) associated to an oligo-
+morphic group G equipped with a measure µ (in a sense that we introduced), satisfying
+certain conditions. Our construction recovers Deligne’s interpolation categories in certain
+cases, and leads to new categories (like the Delannoy category) in other cases. Some con-
+structions and results in [HS1] hold for more general B-categories, and this was our original
+motivation for developing the theory.
+1.7. An application. In forthcoming work [HS3], we give an application of this paper,
+which we now briefly describe. Let C be a pre-Tannakian tensor category. Define Frob(C) to
+be the category whose objects are special commutative Frobenius algebras in C, and whose
+morphisms are co-algebra homomorphisms. We show that Frob(C) is a pre-Galois category,
+and thus (assuming a countability hypothesis) has the form S(G) for some admissible group
+G. We define the oligomorphic component group of C to be the group G. This is an interesting
+invariant of the category C; for example, it recovers the infinite symmetric group from
+Deligne’s category Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep
+Rep(St). Using this, we classify pre-Tannakian categories with enough
+Frobenius algebras, which (we hope) is a step towards a general classification.
+1.8. Outline. In §2, we review oligomorphic and admissible groups and the associated cat-
+egories S(G); these are the motivating examples of pre-Galois categories. In §3, we define
+B-categories and establish some of their basic properties. In §4, we introduce pre-Galois cat-
+egories, and establish some of their special features. In §5, we study the category of atoms
+in a B-category, which leads to the notion of A-category. In §6, we review Fra¨ıss´e theory
+and prove our main theorem. Finally, in §7, we give some examples of A- and B-categories
+coming from relational structures.
+1.9. Notation. We list some of the important notation here:
+0 : the initial object of a B-category (e.g., the empty set)
+1 : the final object of a B-category (e.g., the one-point set)
+S(G) : the category of finitary (and smooth) G-sets
+T(G) : the category of transitive (and smooth) G-sets
+A(B) : the A-category associated to B (see §5.1)
+B(A) : the B-category associated to A (see §5.1)
+2. Oligomorphic groups
+In this section, we review oligomorphic and admissible groups, and recall the category S(G)
+of finitary G-sets. These categories are the motivation for the general notion of pre-Galois
+category we study in this paper.
+2.1. Oligomorphic groups. An oligomorphic group is a permutation group (G, Ω) such
+that G has finitely many orbits on Ωn for all n ≥ 0. Here are a few concrete examples:
+• The infinite symmetric group S, i.e., the group of all permutations of Ω = {1, 2, . . .}.
+• The infinite general linear group over a finite field F, i.e., the group of all linear
+automorphisms of F⊕∞.
+• The group of all order-preserving self-bijections of Q.
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+5
+Many more examples can be obtained from Fra¨ıss´e limits. For example, if R is the Rado
+graph (which is the Fra¨ıss´e limit of all finite graphs) then Aut(R) acts oligomorphically
+on the vertex set of R.
+We refer to Cameron’s book [Cam1] for general background on
+oligomorphic groups.
+2.2. Admissible groups. Fix an oligomorphic group (G, Ω). For a finite subset A of Ω, let
+G(A) be the subgroup of G fixing each element of A. The groups G(A) form a neighborhood
+basis of the identity for a topology on G. This topology has the following properties [HS1,
+§2.2]:
+• It is Hausdorff.
+• It is non-archimedean: open subgroups form a neighborhood basis of the identity.
+• It is Roelcke pre-compact: if U and V are open subgroups then V \G/U is finite.
+We define an admissible group to be a topological group with these three properties. Thus
+every oligomorphic group gives rise to an admissible group. We also note that any finite
+group is admissible (with the discrete topology), and any profinite group is admissible.
+While we are most interested in oligomorphic groups, we typically will not have a preferred
+permutation action, and so it is most natural to work with admissible groups.
+2.3. Actions. Let G be an admissible group. We say that an action of G on a set X is
+smooth if all stabilizers are open. We use the term “G-set” to mean “set equipped with a
+smooth action of G.” We say that a G-set is finitary if it has finitely many orbits. We write
+S(G) for the category of finitary G-sets (with morphisms being G-equivariant maps), and
+T(G) for the full subcategory on the transitive G-sets. An important property of S(G) is
+that it is closed under products and fiber products; see [HS1, §2.3].
+There is a variant of the category S(G) that will play an important role. A stabilizer class
+in G is a collection E of open subgroups of G satisfying the following conditions:
+(a) E contains G.
+(b) E is closed under conjugation.
+(c) E is closed under finite intersections.
+(d) E forms a neighborhood basis for the identity of G.
+We say that a G-set is E -smooth if its stabilizers all belong to E . We let S(G; E ) be the full
+subcategory of S(G) spanned by the E -smooth sets, and analogously define T(G; E ). The
+category S(G; E ) is also closed under products and fiber products.
+Example 2.1. Let S be the infinite symmetric group, let S(n) ⊂ S be the subgroup fixing
+each of 1, . . . , n, and let Sn be the symmetric group on n letters. Let E be the set of all
+subgroups of S conjugate to some S(n), and let Y be the set of all subgroups of S conjugate
+to one of the form Sm1 × · · · Smr × S(n), where m1 + · · · + mr = n. Then E and Y are
+stabilizer classes in S.
+□
+3. Combinatorial tensor categories
+In this section, we introduce the class of B-categories, which we view as combinatorial
+analogs of tensor categories. All categories in this section are essentially small.
+
+6
+NATE HARMAN AND ANDREW SNOWDEN
+3.1. Basic definitions. Let B be a category with finite co-products. We write 0 for the
+initial object and refer to it (or any object isomorphic to it) as empty. We say that an object
+X is atomic, or an atom, if it is non-empty and does not decompose non-trivially under
+co-product; that is, given an isomorphism X ∼= Y ∐ Z either Y or Z is empty.
+We now introduce our combinatorial analog of tensor categories.
+Definition 3.1. A B-category is an essentially small category B satisfying the following
+conditions:
+(a) B has finite co-products.
+(b) Every object of B is isomorphic to a finite co-product of atoms.
+(c) Given objects X, Y , and Z, with X atomic, the natural map
+Hom(X, Y ) ∐ Hom(X, Z) → Hom(X, Y ∐ Z)
+is a bijection.
+(d) B has fiber products and a final object 1.
+(e) Any monomorphism of atoms is an isomorphism.
+We also define a B0-category to be an essentially small category satisfying (a)–(c), and a
+B1-category to be one satisfying (a)–(d).
+□
+The following proposition establishes the motivating example.
+Proposition 3.2. Let G be an admissible group and let E be a stabilizer class. Then the
+category S(G; E ) is a B-category.
+Proof. (a) The co-product is given by disjoint union.
+(b) Atoms are transitive E -smooth G-sets. Every finitary E -smooth G-set is clearly a
+finite disjoint union of transitive E -smooth G-sets.
+(c) Suppose X is an atom and Y and Z are arbitrary objects of S(G; E ). Let f : X → Y ∐Z
+be a map. If any point of X maps into Y (or Z) then all of X maps into Y (or Z) since the
+map is G-equivariant and G acts transitively on X. Thus axiom (c) holds.
+(d) The ordinary fiber product of sets is the fiber product in S(G; E ). The final object is
+the one-point G-set (which is E -smooth since E is required to contain G).
+(e) Suppose f : X → Y is a monomorphism of atoms in S(G; E ). As in any category
+with fiber products, this implies that the projection map X ×Y X → X is an isomorphism.
+Since the set underlying X ×Y X is just the usual fiber product of sets, we see that f is an
+injective function. Since f is an injective map of transitive G-sets, it is bijective, and thus
+an isomorphism in the category.
+□
+Remark 3.3. We mention a few simple ways of producing new B-categories.
+(a) Let B be a B-category and let X be an object of B. Let Σ be the class of all atomic
+objects appearing as a summand of Xn for some n. Let B′ be the full subcategory of
+B spanned by objects that are co-products of objects in Σ. Then B′ is a B-category;
+we call this the subcategory generated by X.
+(b) Let B be a B-category and let S be an object of B. Then the category B/S of objects
+over S is a B-category. If B = S(G) and S = G/U for an open subgroup U then
+B/S = S(U).
+(c) Suppose B1 and B2 are B-categories. Then the product category B1 ⊞ B2 is a B-
+category; we call it the sum category.
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+7
+(d) Let B be a B-category and let 1 = S1 ∐ · · · ∐ Sn be the atomic decomposition of the
+final object. Then B is naturally equivalent to B/S1 ⊞ · · · ⊞ B/Sn, and each B/Si has
+an atomic final object.
+□
+3.2. Properties of B0-categories. Although we are mostly interested in B-categories, some
+results hold in greater generality, and this additional generality is useful in later proofs. In
+this spirit, we now prove some basic results about B0-categories. We fix a B0-category B for
+§3.2.
+Proposition 3.4. If X is non-empty then there are no maps X → 0.
+Proof. It suffices to treat where X is atomic, so we assume this. By Definition 3.1(c) the
+natural map
+Hom(X, 0) ∐ Hom(X, 0) → Hom(X, 0 ∐ 0) = Hom(X, 0)
+is bijective, and so Hom(X, 0) = ∅ as required.
+□
+Proposition 3.5. Let f : X → Y be a morphism. Write X = X1 ∐ · · · ∐ Xn and Y =
+Y1 ∐ · · · ∐ Ym where each Xi and Yi is atomic. There exists a unique function a: [n] → [m]
+such that the restriction of f to Xi factors uniquely through Ya(i); let fi : Xi → Ya(i) be the
+induced map. Then f is uniquely determined by a and the fi’s. Moreover, every choice of a
+and the fi’s comes from some f.
+Proof. For each i, the natural map
+m
+�
+j=1
+Hom(Xi, Yj) → Hom(Xi, Y )
+is a bijection. For m = 0, this is Proposition 3.4, for m = 1 it is obvious, and for m ≥ 2
+it follows from Definition 3.1(c) inductively. We thus see that, given i, there is a unique
+a(i) ∈ [m] and a unique morphism fi : Xi → Ya(j) such that the restriction of f to Xi is fi
+following by the natural map Ya(j) → Y . This proves the existence of a and the fi’s. That
+they determine f, and that every choice arises, follows from the definition of co-product.
+□
+Corollary 3.6. Let f, f ′: X → X′ and g, g′: Y → Y ′ be morphisms. Then f ∐ g = f ′ ∐ g′
+if and only if f = f ′ and g = g′.
+Proposition 3.7. Let f : X → X′ and g : Y → Y ′ be morphisms. Then:
+(a) f ∐ g is monomorphic if and only if f and g are monomorphic.
+(b) f ∐ g is epimorphic if and only if f and g are epimorphic.
+Proof. (a) First suppose that f is not monomorphic. Let h, h′: W → X be distinct maps
+such that fh = fh′. Then h∐idY and h′ ∐idY are maps W ∐Y → X ∐Y , which are distinct
+by Corollary 3.6, but have the same composition with f ∐g. Thus f ∐g is not monomorphic.
+Now suppose that f and g are monomorphic. Let h, h′ : W → X ∐ Y be maps that have
+equal composition with f ∐ g. We show h = h′. It suffices to treat the case where W is
+atomic, since a map out of W is determined by its restrictions to the summands of W. Thus
+assume W is atomic. Then W maps into exactly one of X or Y under h; without loss of
+generality, say X. Then W maps into X′ under (f ∐ g) ◦ h. It follows that W also maps
+into X′ under (f ∐ g) ◦ h′, and so must map into X under h′. Regarding h and h′ as maps
+into X, we thus have (fh ∐ g) = (fh′ ∐ g) as maps W ∐ Y → W ∐ Y ′, and so fh = fh′ by
+Corollary 3.6. Since f is monomorphic, we conclude h = h′. Thus f ∐ g is monomorphic.
+
+8
+NATE HARMAN AND ANDREW SNOWDEN
+(b) First suppose that f is not epimorphic. Let h, h′ : X′ → Z be distinct maps such that
+hf = h′f. Then h ∐ idY ′ and h′ ∐ idY ′ are maps X′ ∐ Y ′ → X ∐ Y ′, which are distinct by
+Corollary 3.6, but have the same composition with f ∐ g. Thus f ∐ g is not epimorphic.
+Now suppose that f and g are epimorphic. Let h, h′ : X′ ∐ Y ′ → Z be maps having equal
+composition with f ∐g. Restricting h and h′ to X′, we see that they have equal composition
+with f. Since f is epimorphic, this means h and h′ have equal restriction to X′. Similarly,
+they have equal restriction to Y ′. By the definition of co-product, this means h = h′, and so
+f ∐ g is epimorphic.
+□
+Corollary 3.8. For any objects X and Y , the natural map X → X ∐Y is a monomorphism.
+Proof. Let i be the identity map of X, and let j : 0 → Y be the unique map. Clearly, i
+and j are monomorphisms. The map in question is (isomorphic to) i ∐ j, and is thus a
+monomorphism by Proposition 3.7(a).
+□
+Proposition 3.9. Fiber products distribute over co-products, in the following sense. Let X,
+X′, and Y be objects of B equipped with morphisms to another object Z. Suppose that the
+fiber products X ×Z Y and X′ ×Z Y exist. Then the fiber product (X ∐ X′) ×Z Y also exists,
+and the natural map
+(X ×Z Y ) ∐ (X′ ×Z Y ) → (X ∐ X′) ×Z Y
+is an isomorphism.
+Proof. Let P = (X ×Z Y ) ∐ (X′ ×Z Y ) and let Φ be the functor on B given by
+Φ(W) =
+�
+Hom(W, X) ∐ Hom(W, X′)
+�
+×Hom(W,Z) Hom(W, Y ).
+Since P has natural maps to X ∐ X′ and Y that agree when composed to Z, there is a
+natural transformation Hom(−, P) → Φ. It suffices to show that this is an isomorphism, for
+then P will represent the fiber product. To check that this is an isomorphism, it suffices to
+verify that Hom(W, P) → Φ(W) is a bijection when W is an atom. In this case, we have
+natural identifications
+Hom(W, P) = Hom(W, X ×Z Y ) ∐ Hom(W, X′ ×Z Y )
+=
+�
+Hom(W, X) ×Hom(W,Z) Hom(W, Y )
+�
+∐
+�
+Hom(W, X′) ×Hom(W,Z) Hom(W, Y )
+�
+=Φ(W),
+and so the result follows.
+□
+3.3. Properties of B-categories. We now prove some general results on B-categories. We
+fix a B-category B for the duration of §3.3.
+Proposition 3.10. The only subobjects of an atom X are 0 and X.
+Proof. Suppose that Y is a non-empty subobject of X. Write Y = Y1 ∐ · · · ∐ Yn with each
+Yi an atom and n ≥ 1. Since Yi → Y is monic by Corollary 3.8, it follows that Yi → X
+is monic, and thus an isomorphism by Definition 3.1(e). It now follows that n = 1, since
+the map X ∐ X → X is not monic (the two natural maps X → X ∐ X are distinct by
+Definition 3.1(c), but have equal composition to X). This completes the proof.
+□
+Proposition 3.11. Any map of atoms is epimorphic.
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+9
+Proof. Let f : X → Y be a map of atoms, and let g, h: Y → Z be maps such that g◦f = h◦f.
+Since B has finite limits, the equalizer Eq(g, h) of g and h exists, and is naturally a subobject
+of Y . Since f factors through Eq(g, h) and X is non-empty, it follows that Eq(g, h) is non-
+empty (Proposition 3.4). Thus Eq(g, h) is equal to Y (Proposition 3.10), and so g = h.
+□
+Proposition 3.12. Let f : X → Y be a morphism. Write X = X1 ∐ · · · ∐ Xn and Y =
+Y1 ∐ · · · ∐ Ym where each Xi and Yi is atomic. Let a: [n] → [m] and fi : Xi → Ya(i) be as in
+Proposition 3.5.
+(a) f is epimorphic if and only if a is surjective.
+(b) f is monomorphic if and only if a is injective and each fi is an isomorphism.
+Proof. For j ∈ [m], let Xj = �
+a(i)=j Xi, and let f j : Xj → Yj be the restriction of f. Then f
+is the co-product of the f j, and so by Proposition 3.7, f is monomorphic (resp. epimorphic)
+if and only if each f j is.
+(a) Suppose that f is monomorphic. Then each f j is monomorphic, and so by Proposi-
+tion 3.10 either Xj is empty or f j is an isomorphism. It follows that a is injective and each
+fi is an isomorphism. Conversely, suppose that a is injective and each fi is an isomorphism.
+Then each f j is clearly monomorphic, and so f is too.
+(b) Suppose that f is epimorphic. Then each f j is epimorphic. It follows that Xj is
+non-empty, as 0 → Yj is not epimorphic (the two maps Yj → Yj ∐ Yj are distinct by
+Definition 3.1(c) but have the same restriction to 0).
+Thus a is surjective.
+Conversely,
+suppose that a is surjective. Then for each j ∈ [m] there is some i with a(i) = j, and then
+map fi is epimorphic by Proposition 3.11. It follows that f j is epimorphic too. Since this
+holds for each j, we find that f is epimorphic.
+□
+Corollary 3.13. The category B is balanced: a morphism that is both monomorphic and
+epimorphic is an isomorphism.
+Proof. Using notation as in the proposition, if f is monomorphic and epimorphic then a is
+a bijection and each fi is an isomorphism, and so f is an isomorphism.
+□
+Corollary 3.14. Let X = X1 ∐ · · · ∐ Xn with each Xi atomic. For a subset S of [n], let
+XS = �
+i∈S Xi. Then every subobject of X is one of the XS, and XS ⊂ XT if and only if
+S ⊂ T.
+Proof. This follows immediately from the structure of monomorphisms given in Proposi-
+tion 3.12.
+□
+Corollary 3.15. Let f : X → Y be a morphism, and use notation as in Proposition 3.5.
+(a) im(f) exists, and is equal to �
+j∈im(a) Yj.
+(b) f is an epimorphism if and only if im(f) = Y .
+(c) The map X → im(f) is an epimorphism, and a monomorphism if and only if f is.
+Proof. This follows from the structure of f given in Proposition 3.5, the characterization of
+monomorphisms and epimorphisms in Proposition 3.12, and the classification of subobjects
+in Corollary 3.14.
+□
+Proposition 3.16. Let f : X → Y be a morphism, and let ∆: X → X ×Y X be the diagonal
+map. The following are equivalent:
+(a) f is monomorphic.
+
+10
+NATE HARMAN AND ANDREW SNOWDEN
+(b) ∆ is an isomorphism.
+(c) ∆ is epimorphic.
+Proof. In any category, (a) and (b) are equivalent, and (b) implies (c). In a balanced category
+(such as a B-category), (c) implies (b) since ∆ is always monomorphic.
+□
+Proposition 3.17. For any objects X and Y , the set Hom(X, Y ) is finite.
+Proof. Consider a map f : X → Y . Let Γf ⊂ X × Y be the image of idX × f : X → X × Y ,
+and let p: Γf → X and q: Γf → Y be the projections. Since idX × f is a monomorphism,
+it follows from Corollary 3.15 that the natural map X → Γf is both a monomorphism and
+an epimorphism, and is thus an isomorphism by Corollary 3.13; its inverse is clearly p. We
+thus see that f = q ◦ p−1, and so f can be recovered from Γf. As X × Y has only finitely
+many subobjects (by Corollary 3.14), the result follows.
+□
+Corollary 3.18. Any self-map of an atom is an isomorphism.
+Proof. Let f : X → X be a map with X an atom. Then f is an epimorphism (Proposi-
+tion 3.11), and so f ∗: Hom(X, X) → Hom(X, X) is injective. Since Hom(X, X) is finite
+(Proposition 3.17), it follows that f ∗ is bijective, and so there exists g ∈ Hom(X, X) such
+g ◦ f = idX. Thus f is a monomorphism, and hence an isomorphism (Corollary 3.13).
+□
+3.4. Orbits. Suppose G is an admissible group and X is a finitary G-set. One can then
+form the orbit space G\X, which is a finite set. Passing to orbits is often an important idea.
+There is an analog of this construction in our more general categories. Let B be a B0-
+category. We define the orbit set of X, denoted Xorb, to be the set of atomic subobjects
+of X. This construction is natural: it follows from Proposition 3.5 that a map f : X → Y
+naturally induces a function f orb: Xorb → Y orb. We therefore have a functor
+B → FinSet,
+X �→ Xorb,
+where FinSet is the category of finite sets.
+We now show how one can read off some
+properties of a morphism from how it behaves on orbits.
+Proposition 3.19. Suppose B is a B-category and f : X → Y is a morphism.
+(a) f is epimorphic if and only if f orb is surjective.
+(b) f is monomorphic if and only if Xorb → (X ×Y X)orb is surjective (or bijective); in
+this case, f orb is injective.
+Proof. (a) follows from Proposition 3.12(a).
+We now prove (b).
+Let ∆: X → X ×Y X
+be the diagonal. If f is monomorphic then ∆ is an isomorphism (Proposition 3.16), and
+so ∆orb is a bijection; conversely, if ∆orb is surjective then ∆ is epimorphic by (a), and
+so f is monomorphic (Proposition 3.16).
+If f is monomorphic then f orb is injective by
+Proposition 3.12(b).
+□
+Remark 3.20. Let B be a B1-category. One can sometimes modify B to produce a B-
+category, as we now describe. Let f : X → Y be a morphism in B. We make the following
+definitions:
+• f is a pre-monomorphism if the map Xorb → (X ×Y X)orb is bijective.
+• f is a pre-epimorphism if the map Xorb → Y orb is surjective.
+• f is a pre-isomorphism if it is a pre-monomorphism and pre-isomorphism.
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+11
+Suppose that the class of pre-isomorphisms is stable under base change. Then this class
+forms a right multiplicative system, as defined in [Stacks, Tag 04VC]. The localized category
+is a B-category, and is the universal B-category to which B maps (with respect to functors
+that preserve finite co-products, finite limits, and atoms).
+□
+4. Pre-Galois categories
+In this section, we identify a few categorical properties of S(G) that need not hold for
+a general B-category, the most important of which is non-degeneracy. Motivated by these
+observations, we introduce the class of pre-Galois categories. We also discuss how they relate
+to the existing notion of Galois category. All categories in this section are assumed to be
+essentially small.
+4.1. Non-degeneracy. We begin with the following observation.
+Proposition 4.1. Let B be a B1-category. The following are equivalent:
+(a) If X → Z and Y → Z are maps of atoms then X ×Z Y is non-empty.
+(b) A base change of an epimorphism is an epimorphism.
+(c) A product of epimorphisms is an epimorphism.
+Proof. (a) ⇒ (b). Let f : X → Y be an epimorphism, let Y ′ → Y be an arbitrary map, and
+let f ′: X′ → Y ′ be the base change of f. We show that f ′ is an epimorphism. Since fiber
+products distribute over co-products, it suffices to treat the case where X, Y , and Y ′ are
+atoms. By assumption, X′ is then non-empty, and so f ′ is an epimorphism.
+(b) ⇒ (c). Let X → Y and X′ → Y ′ be epimorphisms. Consider the composition
+X × X′ → Y × X′ → Y × Y ′.
+The first map is the base change of the epimorphism X → Y along the map X′ → 1, and
+is thus an epimorphism; similarly, the second map is the base change of the epimorphism
+X′ → Y ′ along the map Y → 1, and is thus an epimorphism. It follows that the composition
+X × X′ → Y × Y ′ is an epimorphism, as required.
+(c) ⇒ (a). Let X → Y and Y ′ → Y be maps of atoms, and let X′ = X ×Y Y ′ be the fiber
+product. Since X → Y is an epimorphism, by assumption X′ → Y ′ is also an epimorphism.
+Thus X′ is non-empty.
+□
+Motivated by the above proposition, we make the following definition.
+Definition 4.2. A B1-category is non-degenerate if the equivalent conditions of Proposi-
+tion 4.1 hold, and the final object 1 is atomic.
+□
+It is clear that the category S(G; E ) is non-degenerate, for any admissible group G and
+stabilizer class E . In Example 7.3, we give an interesting example of a degenerate B-category.
+4.2. Implications of non-degeneracy. Fix a non-degenerate B-category B. We now ex-
+amine some consequences of the non-degeneracy condition. We note that these results can
+be deduced from the classification of such categories (provided by Theorem 6.15), but we
+find it instructive to give direct proofs. For a morphism f : X → Y , we define the kernel
+pair of f to be Eq(f) = X ×Y X. It is a subobject of X × X.
+Proposition 4.3. Let f : X → Y and g : X → Z be epimorphisms. Then f factors through
+g if and only if Eq(g) ⊂ Eq(f).
+
+12
+NATE HARMAN AND ANDREW SNOWDEN
+Proof. It is clear that if f factors through g then Eq(g) ⊂ Eq(f). We now prove the converse;
+thus assume Eq(g) ⊂ Eq(f). Let I be the image of X in Y ×Z, and let h: X → I, p: I → Y ,
+and q: I → Z be the natural maps; note that f = p ◦ h and g = q ◦ h. We have
+Eq(h) = Eq(f × g) = Eq(f) ∩ Eq(g) = Eq(g),
+where f × g denotes the map X → Y × Z. Consider the commutative diagram
+X ×I X
+�
+�
+X ×Z X
+�
+I
+� I ×Z I
+The top map is the inclusion Eq(h) ⊂ Eq(g), which is an isomorphism. The right map is
+an epimorphism since h is an epimorphism and the category B is non-degenerate; to be a
+little more precise, note that this morphism is the base change of X × X → I × I along
+the diagonal Z → Z × Z. It follows that the bottom map is an epimorphism, and so q is
+an monomorphism (Proposition 3.16), and thus an isomorphism (Corollary 3.13). We thus
+have f = p ◦ q−1 ◦ g, which completes the proof.
+□
+Corollary 4.4. For X fixed, there are finitely many epimorphisms X → Y up to isomor-
+phism.
+Proof. By the proposition, an epimorphism f : X → Y is determined up to isomorphism
+by Eq(f), which is a subobject of X × X.
+Since X × X has finitely many subobjects
+(Corollary 3.14), the result follows.
+□
+Proposition 4.5. A non-degenerate B-category B is finitely co-complete.
+Proof. Since B has finite co-products, it suffices to show that it has co-equalizers.
+Let
+f, g : X → Y be parallel morphisms. Let {qi : Y → Zi}i∈U be representatives of the isomor-
+phism classes of epimorphisms out of Y ; this set is finite by Corollary 4.4. Let V be the set
+of indices i ∈ U such that qi ◦ f = qi ◦ g. Define I to be the image of the map Y → �
+i∈V Zi,
+and let h: Y → I be the natural map. We claim that h is a co-equalizer of (f, g).
+To see this, suppose that a: Y → T is a morphism with a ◦ f = a ◦ g. The morphism a
+factors as c◦b, where b is an epimorphism and c is a monomorphism; we may as well assume
+b = qi for some i ∈ U. Since c is a monomorphism, it follows that qi ◦ f = qi ◦ g, and so
+i ∈ V . Let pi : I → Zi be the projection onto the ith factor, so that qi = pi ◦ h. Composing
+with c, we have a = c ◦ pi ◦ h. We thus see that a factors through h. The factorization is
+unique since h is an epimorphism.
+□
+Remark 4.6. The above proof actually shows that any B-category satisfying Corollary 4.4
+is finitely co-complete. All B-categories we know (including the degenerate ones) satisfy this
+corollary.
+□
+4.3. Effective equivalence relations. Let G be an admissible group and let E be a stabi-
+lizer class. By Proposition 4.5, the category S(G; E ) is finitely co-complete. This is somewhat
+surprising, since every smooth G-set is a quotient of some E -smooth G-set. The explanation
+here is that co-equalizers in S(G; E ) do not agree with co-equalizers in S(G). In fact, S(G; E )
+is a reflective subcategory of S(G), and co-equalizers in S(G; E ) are obtained by computing
+in S(G) and then applying the reflector. We now give an example to illustrate the situation.
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+13
+Example 4.7. Let G = S be the infinite symmetric group acting on Ω = {1, 2, . . .}. Let
+Ω[2] be the subset of Ω2 consisting of pairs (x, y) with x ̸= y and let Ω(2) be the set of
+2-element subsets of Ω. Let p: Ω[2] → Ω(2) be the natural surjection, and let R = Eq(p) be
+the kernel-pair of p. In the category S(G), the co-equalizer of R ⇒ Ω[2] is Ω(2).
+Now, let E be the stabilizer class consisting of subgroups conjugate to some S(n), as
+in Example 2.1.
+The G-sets Ω[2] and R are E -smooth, while Ω(2) is not.
+The reflector
+Φ: S(G) → S(G; E ) is computed on transitive G-sets by Φ(G/U) = G/V , where V is the
+minimal open subgroup over U that belongs to E (it is not difficult to see directly that such
+a subgroup exists). We have Ω(2) ∼= G/U, where U = S2 × S(2). From the classification of
+open subgroups of S (see, e.g., [HS1, Proposition 15.1]), we see that the only subgroup in E
+containing U is S itself. Thus Φ(Ω(2)) = 1 is the one-point set, and this is the co-equalizer
+of R ⇒ Ω[2] in the category S(S; E ).
+□
+The following terminology is useful for explaining this situation:
+Definition 4.8. Let C be a finitely complete category. We say that an equivalence relation
+R on an object X is effective if the quotient X/R exists (this is defined as the co-equalizer
+of R ⇒ X), and the kernel pair of the quotient map X → X/R is R itself. We say that C
+has effective equivalence relations if all equivalence relations in C are effective.
+□
+With this terminology, Example 4.7 can be summarized as follows: R is an effective
+equivalence relation in S(S), but not in the subcategory S(S; E ). The following proposition
+gives the general statement in this direction.
+Proposition 4.9. Let G be an admissible group and let E be a stabilizer class.
+(a) The category S(G) has effective equivalence relations.
+(b) If S(G; E ) has effective equivalence relations then S(G; E ) = S(G), i.e., E contains
+all open subgroups of G.
+Proof. (a) The category of sets has effective equivalence relations. This property passes to
+S(G) since finite limits and co-limits here are computed on the underlying sets.
+(b) Let U be an open subgroup of G, and let V be a member of E with V ⊂ U. Put
+Y = G/V and X = G/U, let π: Y → X be the natural map, and let R ⊂ Y × Y be the
+kernel pair of π. Since Y × Y belongs to S(G; E ), so does the subobject R, and so R defines
+an equivalence relation on Y in the category S(G; E ). Thus, by assumption, there is a map
+π′: Y → X′ in S(G; E ) with kernel pair R; of course, we may as well assume π′ is surjective.
+Since the inclusion of S(G; E ) into S(G) preserves fiber products, it follows that R is the
+kernel pair of π′ in S(G). Thus π and π′ are isomorphic. In particular, G/V is E -smooth,
+and so V belongs to E .
+□
+4.4. Pre-Galois categories. We now introduce this class of categories:
+Definition 4.10. A pre-Galois category is a non-degenerate B-category with effective equiv-
+alence relations.
+□
+This definition is equivalent to the one given in the introduction. As the preceding dis-
+cussion shows, if G is an admissible group then S(G) is a pre-Galois category.
+4.5. Comparison with Galois categories. We now discuss the relation between the clas-
+sical notion of Galois category and our notion of pre-Galois category. We begin by recalling
+the former:
+
+14
+NATE HARMAN AND ANDREW SNOWDEN
+Definition 4.11. A Galois category is a pair (C, ω) where C is a category and ω : C → FinSet
+is a functor (the fiber functor) such that the following axioms hold:
+(a) C has finite limits and colimits.
+(b) Every morphism X → Y in C factors as X → I → Y , where I is a summand of Y
+and X → I is a strict epimorphim, i.e., X → I is the co-equalizer of X ×I X ⇒ X.
+(c) ω is exact, i.e., it commutes with finite limits and co-limits.
+(d) ω is conservative, i.e., ω(ϕ) is an isomorphism if and only if ϕ is.
+We note there are other axiomizations; this one comes from [Cad, §2.1.1].
+□
+The following is the main result we are after.
+Proposition 4.12. Let B be a category and ω : B → FinSet a functor. The following are
+equivalent:
+(i) (B, ω) is a Galois category.
+(ii) B is a pre-Galois category and ω is exact and conservative.
+Proof. Suppose (i) holds. By the main theorem of Galois categories [Cad, Theorem 2.8],
+up to equivalence, B is the category of finite G-sets, for some pro-finite group G, and ω is
+the forgetful functor. Since G is an admissible group and B = S(G), it follows that B is
+pre-Galois. Thus (ii) holds.
+Now suppose (ii) holds. We verify the conditions of Definition 4.11. Conditions (c) and (d)
+hold by assumtion. Any B-category is finitely complete by definition, and a non-degenerate
+one is finitely co-complete by Proposition 4.5; thus (a) holds. Every morphism f in a B-
+category factors as f = g◦h, where h is an epimorphism and g is the inclusion of a summand.
+Thus to complete the proof of (b), it suffices to show that every epimorphism is strict.
+Let f : X → Y be an epimorphism, and let R = Eq(f) be its kernel pair. Since equivalence
+relations are effective, the quotient g : X → X/R exists, and R = Eq(g). By Proposition 4.3,
+we see that g and f are isomorphic. Since g is the co-kernel of R, so is f, i.e., f is strict.
+□
+The proposition can be summarized as: “Galois = pre-Galois + fiber functor.”
+5. Categories of atoms
+A B-category is completely determined by its atoms. In this section, we make this state-
+ment precise: we introduce the notion of an A-category, and show that A-categories are
+exactly the (opposite) categories of atoms in a B-categories. The A-category perspective is
+useful since it provides a bridge between B-categories and finite relational structures. All
+categories in this section are assumed to be essentially small.
+5.1. The A and B constructions. Let B be a B0-category. We define A(B) to be the full
+subcategory of Bop spanned by the atoms of B. For example, if B = S(G) then A(B) =
+T(G)op is the opposite of the category of transitive G-sets.
+Let A be an essentially small category. We define a category B(A) as follows. An object
+of B(A) is a finite sequence X• = (X1, . . . , Xn) where Xi is an object of A. A morphism
+(X1, . . . , Xn) → (Y1, . . . , Ym) consists of a function a: [n] → [m] together with a morphism
+Xi → Ya(i) in Aop for each i ∈ [n]. Composition is defined in the obvious manner.
+Proposition 5.1. For any B0-category B, we have an equivalence Φ: B(A(B)) → B given
+on objects by
+Φ((X1, . . . , Xn)) = X1 ∐ · · · ∐ Xn.
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+15
+Proof. This follows from the basic properties of B0-categories established in §3.2.
+□
+Proposition 5.2. For any category A, the category B(A) is a B0-category and we have a
+natural equivalence A ∼= A(B(A)).
+Proof. (a) It is clear that co-products in B(A) are given on objects by
+(X1, . . . , Xm) ∐ (Y1, . . . , Yn) = (X1, . . . , Xm, Y1, . . . , Yn),
+with the obvious structure maps. We note that the zero object of B(A) is the empty sequence
+().
+(b) Suppose that X• = (X1, . . . , Xn) and Y• = (Y1, . . . , Ym) are isomorphic objects of
+B(A). Let (a, f): X• → Y• be the given isomorphism, where a: [n] → [m] is a map of sets
+and fi : Xi → Ya(i) is a morphism in Aop, and let (b, g): Y• → X• be its inverse. Since the
+composition is the identity, it follows that b ◦ a and a ◦ b are the identity maps of [n] and
+[m]; thus n = m and a and b are inverse permutations. Moreover, fi : Xi → Ya(i) is an
+isomorphism with inverse ga(i).
+From the above, together with the description of the co-product on B(A), it follows that
+(X) is an atomic object of B(A), for any object X of A. We thus see that every object of
+B(A) is a finite co-product of atomic objects.
+(c) It follows from the definition of morphisms in B(A) that the natural map
+HomB(A)((X), Y• ∐ Z•) → HomB(A)((X), Y•) ∐ HomB(A)((X), Z•)
+is bijective, for any object X of A and objects Y• and Z• of B(A).
+□
+We thus see that there is a correspondence between B0-categories and all (essentially small)
+categories. In the remainder of this section, we refine this correspondence, and determine
+what B1- and B-categories correspond to. To this end, we begin with one simple observation:
+Proposition 5.3. Let f : X → Y be a morphism in the category A, and let f ′: (Y ) → (X)
+be the corresponding morphism in B(A). Then f is an isomorphism (resp. monomorphism,
+epimorphism) if and only if f ′ is an isomorphism (resp. epimorphism, monomorphism).
+Proof. The statement for isomorphisms is clear, as an inverse to one of f or f ′ gives an
+inverse to the other. It is also clear that if f is not a monomorphism then f ′ is not an
+epimorphism, as a witness to the failure of the former leads to one for the latter. Similarly,
+it is clear that if f is not an epimorphism then f ′ is not a monomorphism.
+Now suppose that f ′ is not a monomorphism.
+Then there exist distinct morphisms
+g′, h′: (Z1, . . . , Zn) → (Y ) such that f ′ ◦ g′ = f ′ ◦ h′. Let g′
+i and h′
+i be the components
+of g′ and h′, and let gi and hi be the corresponding morphisms in A. Since g′ ̸= h′ there is
+some i such that g′
+i ̸= h′
+i. Thus gi and hi are distinct morphisms in A with gi ◦ f = hi ◦ f,
+and so f is not an epimorphism.
+Finally, suppose that f ′ is not an epimorphism.
+Then there exist distinct morphisms
+g′, h′: (X) → (W1, . . . , Wn) such that g′ ◦ f ′ = h′ ◦ f ′. By definition, g′ corresponds to a
+morphism g : Wi → X for some i, and h′ to a morphism h: Wj → X for some j. The equality
+g′ ◦ f ′ = h′ ◦ f ′ exactly means that i = j and g ◦ f = h ◦ f. Since g′ ̸= h′ we have g ̸= h, and
+so f is not a monomorphism.
+□
+5.2. Initial objects. Let A be a category. We say that a set S of objects of A is an initial
+set if for every object X of A there exists a unique object I of S such that HomA(I, X)
+is non-empty, and this set contains a single element. Suppose A has an initial set S. For
+
+16
+NATE HARMAN AND ANDREW SNOWDEN
+I ∈ S, let AI be the full subcategory of A spanned by objects X for which there exists a
+map I → X. Then AI has I as an initial object, and A is the disjoint union of the AI’s
+(as a category). Conversely, if A is a (set-indexed) disjoint union of categories with initial
+objects, then A has an initial set.
+Proposition 5.4. Let A be a category, and let B = B(A).
+(a) A has a finite initial set if and only if B has a final object.
+(b) A has an initial object if and only if B has an atomic final object.
+Proof. (a) Suppose that {I1, . . . , In} is an initial object of A. We claim that I• = (I1, . . . , In)
+is a final object of B. Indeed, let X• = (X1, . . . , Xm) be given. For each 1 ≤ i ≤ m there is
+a unique 1 ≤ a(i) ≤ n such HomA(Ia(i), Xi) is non-empty, and it contains a single element
+fi. The map a together with f1, . . . , fn define a morphism X• → I• in B, and it is clearly
+the unique such map. Thus I• is a final object of B. This reasoning is reversible too: if I•
+is a final object of B then {I1, . . . , In} is an initial set of A.
+(b) This is clear from the proof of (a).
+□
+5.3. Amalgamations. A pre-amalgamation in A is a pair of morphisms (b: A → B, c: A →
+C). Given a pre-amalgamation (b, c), define Amalg(b, c) to be the category whose objects are
+pairs (b′ : B → D, c′: C → D) of morphisms in A with b′b = c′c, with the obvious morphisms.
+An amalgamation set for (b, c) is an initial set of this category; we call the elements of this
+set amalgamations.
+Proposition 5.5. Let A be a category and let B = B(A) be the corresponding B0-category.
+The following are equivalent:
+(a) Every pre-amalgamation in A has a finite amalgamation set.
+(b) The category B has fiber products.
+Proof. Suppose (b) holds. Let (b, c) be a pre-amalgamation in A, where b: A → B and
+c: A → C. Let (X1, . . . , Xn) be the fiber product of (B) with (C) over (A) in B. The
+map (X1, . . . , Xn) → (B) in B corresponds to morphisms fi : B → Xi in A, for 1 ≤ i ≤ n.
+Similarly, the map (X1, . . . , Xn) → C corresponds to morphisms gi : Xi → C in A, for
+1 ≤ i ≤ n. Clearly, fi ◦ a = gi ◦ b, so each (fi, gi) is an object of Amalg(b, c).
+We claim that S = {(fi, gi)}1≤i≤n is an amalgamation set for (b, c). Thus let (f : B →
+Y, g : C → Y ) be an arbitrary object of Amalg(b, c). Then f defines a morphism (Y ) → (B)
+in B, and similarly, g defines a morphism (Y ) → (C) in B. The two composition to (A) agree,
+and so there is a unique morphism (Y ) → (X1, . . . , Xn) that composes with the projections
+to the given morphisms. This proves the claim, and so (a) holds.
+Now suppose (a) holds. Let (B) → (A) and (C) → (A) be morphisms of atoms in B,
+corresponding to maps b: A → B and c: A → C in A. Let {(fi, gi)}1≤i≤n be an amalgamation
+set for (b, c), where fi and gi map to Xi. Then, reversing the above reasoning, we see that
+(X1, . . . , Xn) is naturally the fiber product of (B) and (C) over (A).
+We thus find that the fiber product of morphisms of atoms in B always exists. It follows
+from Proposition 3.9 that all fibers products exist.
+□
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+17
+We say that a category A has the amalgamation property (AP) if for every pre-amalgamation
+(b, c) the category Amalg(b, c) is non-empty. This means that every diagram
+B
+� D
+A
+b
+�
+c
+� C
+�
+can be filled, i.e., one can find D and the dotted arrows making the square commute.
+Proposition 5.6. Let A be a category in which all pre-amalgamations have a finite amal-
+gamation set, and let B = B(A). Then A has the amalgamation property if and only if for
+every morphism of atoms X → Z and Y → Z in B, the fiber product X ×Z Y is non-empty.
+Proof. This is clear from the proof of Proposition 5.5.
+□
+5.4. A-categories. We are finally ready to introduce the main concept of this section:
+Definition 5.7. An A-category is an essentially small category A satisfying the following
+conditions:
+(a) The category A has a finite initial set.
+(b) Every pre-amalgamation has a finite amalgamation set.
+(c) Every epimorphism in A is an isomorphism.
+An A1-category is an essentially small category satisfying conditions (a) and (b).
+□
+The following is the main result of this section:
+Theorem 5.8. Let A be a category and put B = B(A).
+(a) B is a B1-category ⇐⇒ A is an A1-category.
+(b) B is a B-category ⇐⇒ A is an A-category.
+(c) B is a non-degenerate B-category ⇐⇒ A is an A-category with an initial object and
+the amalgamation property.
+Proof. (a) follows from Propositions 5.2, 5.4(a), and 5.5; (b) then follows from Proposi-
+tion 5.3; and (c) then follows from Propositions 5.4(b) and 5.6.
+□
+Corollary 5.9. All morphisms in an A-category are monomorphisms.
+Proof. This follows from Propositions 3.11 and 5.3.
+□
+Corollary 5.10. Any endomorphism in an A-category is an isomorphism.
+Proof. This follows from Corollary 3.18 and 5.3.
+□
+Remark 5.11. A category in which all endomorphisms are isomorphisms is called an EI-
+category. Thus the above corollary shows that every A-category is an EI-category. Repre-
+sentations of EI-categories have received some attention in the literature, e.g., [GL].
+□
+We now discuss the condition Definition 5.7(c) in a bit more detail. The contrapositive of
+Definition 5.7(c) can be phrased as follows: if f : X → Y is a non-isomorphism then there
+exist distinct morphisms g1, g2: Y → Z such that g1 ◦ f = g2 ◦ f. As Corollary 5.9 suggests,
+when working on the “A side,” morphisms will in some sense be embeddings. From this
+perspective, Definition 5.7(c) essentially means that if X is a proper subobject of Y then we
+can find distinct embeddings of Y into some auxiliary object that agree on X.
+
+18
+NATE HARMAN AND ANDREW SNOWDEN
+There is one other perspective on Definition 5.7(c) that is sometimes useful. Let f : X → Y
+be a morphism in an A1-category. We refer to objects in the amalgamation set of (f, f) as
+self-amalgamations of Y over X. There is always a trivial self-amalgamation, namely Y
+itself, or more precisely, the pair (idY , idY ). One easily sees that f is an epimorphism if and
+only if this is the only self-amalgamation. Thus the contrapositive of Definition 5.7(c) is
+equivalent to the following: if f : X → Y is a non-isomorphism then there is a non-trivial
+self-amalgamation of Y over X.
+5.5. Products. Let A1 and A2 be A1-categories. One easily sees that the product category
+A1 × A2 is also an A1-category, and is an A-category if both A1 and A2 are. This motivates
+the following construction:
+Definition 5.12. Let B1 and B2 be B1-categories. We define the tensor product category
+to be the B1-category
+B1 ⊠ B2 = B(A(B1) × A(B2)).
+If B1 and B2 are both B-categories then so is B1 ⊠ B2.
+□
+Example 5.13. Let G1 and G2 be admissible groups with stabilizer classes E1 and E2. One
+can then show
+S(G1; E1) ⊠ S(G2; E2) ∼= S(G1 × G2; E1 × E2),
+where E1 × E2 denotes the set of open subgroups of the product of the form U1 × U2 with
+Ui ∈ Ei. Note that if E1 and E2 each contain all open subgroups then the same need not be
+true for E1 ×E2. Thus one is essentially forced to confront stabilizer classes when considering
+the tensor product construction.
+□
+6. Fra¨ıss´e theory
+In this section, we review classical Fra¨ıss´e theory and its categorical reformulation, and
+then apply this theory to prove the main theorems of this paper.
+6.1. Classical Fra¨ıss´e theory. We now recall the classical formulation Fra¨ıss´e’s theorem.
+While we will not apply this version of the theorem, it serves as motivation for the categorical
+form discussed in §6.2 that we do use. We will also use the language of relational structures
+in §7 to construct examples of A-categories. We refer to [Cam1] and [Mac] for more complete
+discussions.
+A signature is a collection Σ = {(Ri, ni)}i∈I where Ri is a formal symbol and ni is a
+positive integer, called the arity of Ri. Fix a signature Σ. A (relational) structure for Σ
+is a set X equipped with for each i ∈ I an ni-ary relation Ri on X (i.e., a subset of Xni).
+Given a structure X and a subset Y , there is an induced structure on Y ; we call structures
+obtained in this manner substructures of X. An embedding of structures X → Y is an
+injective function that identifies X with a substructure of Y .
+A structure Ω is called homogeneous if whenever X and Y are finite substructures and
+i: X → Y is an isomorphism of structures, there exists an automorphism σ of Ω such that
+σ(x) = i(x) for all x ∈ X. The age of a structure Ω, denoted age(Ω), is the set of all finite
+structures that embed into Ω. If Ω is a countable homogeneous structure then C = age(Ω)
+has the following properties:
+• C is hereditary: if Y belongs to C and X is (isomorphic to) a substructure of Y then
+X belongs to C.
+• The set |C| of isomorphism classes in C is countable.
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+19
+• C satisfies the amalgamation property, as defined in §5.3; here we treat C as a category
+with morphisms being embeddings.
+Fra¨ıss´e’s theorem is the converse statement: if C is a class of finite structures satisfying the
+above three conditions then C is the age of a countable homogeneous structure Ω, which is
+unique up to isomorphism. A class satisfying the above conditions is called a Fra¨ıss´e class,
+and the resulting homogeneous structure Ω is called the Fra¨ıss´e limit of C.
+For a class C of structures, let Cn denote the subclass consisting of structures with n
+elements. Suppose Ω is a homogeneous structure and C = age(Ω) has the property that |Cn|
+is finite for all n. Then one easily sees that G = Aut(Ω) acts oligomorphically on Ω. In this
+way, Fra¨ıss´e limits provide a powerful mechanism for constructing oligomorphic groups.
+Example 6.1. We give a few examples of Fra¨ıss´e limits.
+(a) Take the signature to be empty, so that a structure is simply a set. The class C of all
+finite sets is a Fra¨ıss´e class, and the Fra¨ıss´e limit Ω is a countable infinite set. The
+oligomorphic group G = Aut(Ω) is the infinite symmetric group.
+(b) Take the signature to consist of a single binary relation. The class C of all finite
+totally ordered sets is a Fra¨ıss´e class, and the Fra¨ıss´e limit Ω is the set of rational
+numbers equipped with its standard total order.
+(c) Again, take the signature to consist of a single binary relation. Let C be the class of
+all finite simple graphs. This is a Fra¨ıss´e class, and the limit is the Rado graph.
+□
+6.2. Categorical Fra¨ıss´e theory. Given a class C of relational structures, one can regard C
+as a category with morphisms being embeddings. Fra¨ıss´e’s theorem is thus a statement about
+a certain class of categories. It turns out that the theorem actually holds for a much broader
+class of categories. This observation goes back to the work of Droste–G¨obel [DG1, DG2],
+and has been discussed in more recent work as well [Car, Irw, Kub]. We follow the treatment
+in the appendix to our recent paper [HS2].
+Fix a category C in which all objects are monomorphisms; we often refer to morphisms
+in C as embeddings. An ind-object in C is a diagram X1 → X2 → · · · in C. It is possible
+to consider ind-objects indexed by more general posets, but we will only need this simple
+version. There is a natural notion of morphism between ind-objects, and between an ordinary
+object and an ind-object; see [HS2, §A.2].
+Let Ω be an ind-object of C. We say that Ω is universal if every object of C embeds into
+Ω. We say that Ω is homogeneous if every isomorphism of finite subobjects is induced by an
+automorphism. Precisely, this means the following. Suppose α: X → Ω and β : Y → Ω are
+embeddings, where X and Y are objects of C, and that we have an isomorphism γ : X → Y
+in C. Then there must exist an automorphism σ of Ω such that σ ◦ α = β ◦ γ. We say that
+C is a Fra¨ıss´e category if it admits a universal homogeneous ind-object. We note that any
+two universal homogeneous ind-objects are isomorphic [HS2, Proposition A.7].
+Fra¨ıss´e’s theorem gives a characterization of Fra¨ıss´e categories. To state it, we will need
+the amalgamation property (AP) defined in §5.3, as well as the following condition:
+(RCC) Relative countable cofinality: for any object X of C there exists a cofinal sequence of
+morphisms out of X, i.e., there is a sequence of morphisms {αn : X → Yn}n≥1 such
+that if β : X → Y is any morphism then there is a morphism γ : Y → Yn for some
+n such that γ ◦ β = αn.
+The following is the categorical Fra¨ıss´e theorem (in one form).
+
+20
+NATE HARMAN AND ANDREW SNOWDEN
+Theorem 6.2 ([HS2, Theorem A.11]). Suppose that C has an initial object. Then C is a
+Fra¨ıss´e category if and only if (RCC) and (AP) hold.
+Example 6.3. Here is an example where the categorical Fra¨ıss´e theorem applies while the
+classical one does not apply. A cubic space is a complex vector space V equipped with a
+linear map Sym3(V ) → C. There is a natural notion of embedding for cubic spaces. In
+[HS2], we show that the category of finite dimensional cubic spaces is a Fra¨ıss´e category; we
+give many other related examples as well.
+□
+6.3. Fra¨ıss´e theory for A-categories. The following is our main Fra¨ıss´e-like theorem for
+A-categories.
+Theorem 6.4. Let A be an A-category satisfying the following conditions:
+• A has an initial object.
+• A satisfies the amalgamation property.
+• A has countably many isomorphism classes.
+Then there exists an admissible group G and a stabilizer class E for G such that A is
+equivalent to T(G; E )op.
+We will actually prove a slightly more precise statement. Let A be any category satisfying
+the three conditions of Theorem 6.4. By Theorem 6.2, the category A is Fra¨ıss´e, and thus
+admits a universal homogeneous ind-object Ω. Let G be its automorphism group. For an
+object X, we let Φ(X) be the set of all embeddings X → Ω; note that this is non-empty
+since Ω is universal. The group G naturally acts on Φ(X), via its action on Ω, and this
+action is transitive by homogeneity. Give α ∈ Φ(X), we let Gα be the stabilizer of α in G.
+Let E be the set of all subgroups of G of the form Gα, for some α.
+Theorem 6.5. Let A be an A1-category satisfying the three conditions of Theorem 6.4, and
+let Ω, G, E , and Φ be as above.
+(a) The family E is a neighborhood basis for a first-countable admissible topology on G.
+(b) The family E is a stabilizer class for G.
+(c) The construction Φ defines a faithful and essentially surjective functor A → T(G; E )op.
+(d) If A is an A-category then the functor in (c) is an equivalence.
+Remark 6.6. In §7.4, we give an example of an A1-category (that is not an A-category)
+where the functor in (c) is not an equivalence.
+□
+Remark 6.7. There is a notion of completeness for admissible groups. In Theorem 6.4, there
+is in fact a unique (up to isomorphism) complete group satisfying the concluding statement.
+The group G constructed following the statement of the theorem is thie complete group.
+□
+We now prove the theorem, in a series of lemmas. We fix A, Ω, G, E , and Φ as in the
+theorem statement in what follows. We also write 1 for the initial object of A.
+Lemma 6.8. Let X and Y be objects of A, and let α: X → Ω and β : Y → Ω be embeddings.
+Then there is a unique (up to isomorphism) diagram
+Y
+δ
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+β
+�
+1
+�♠
+♠
+♠
+♠
+♠
+♠
+♠
+♠
+♠
+♠
+♠
+�◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+Z
+ǫ
+� Ω
+X
+γ
+�❧
+❧
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+α
+�
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+21
+where (Z, γ, δ) is an amalgamation of X and Y over the trivial object 1. We have Gǫ =
+Gα ∩ Gβ.
+Proof. The existence and uniqueness of the diagram follow from the definition of A1-category.
+We have α = ǫγ, and so for σ ∈ G we have σα = σǫγ; thus Gǫ ⊂ Gα. Of course, the same
+holds with β, and so Gǫ ⊂ Gα ∩ Gβ. We now prove the reverse containment. Thus let
+σ ∈ Gα ∩ Gβ be given.
+Then the above diagram commutes with ǫ changed to σǫ.
+By
+uniqueness of the above diagram, it follows that ǫ = σǫ, and so σ ∈ Gǫ, as required.
+□
+Lemma 6.9. Let X and Y be objects of A, let E = G\(Φ(X) × Φ(Y )), and let F be
+an amalgamation set for X and Y over 1. Then we have a natural bijection E ∼= F; in
+particular, E is finite.
+Proof. Given α ∈ Φ(X) and β ∈ Φ(Y ), let (Z, γ, δ) be the amalgamation from Lemma 6.8. It
+is clear that if (α, β) is modified by an element of G then the amalgamation is unchanged (up
+to isomorphism). This construction therefore yields a well-defined map E → F. Conversely,
+if (Z, γ, δ) is any amalgamation then by choosing an embedding ǫ: Z → Ω, we get the pair
+(γ∗(ǫ), δ∗(ǫ)) in Φ(X) × Φ(Y ), and the orbit of this pair is independent of the choice fo ǫ.
+This provides a map F → E. One readily verifies the two maps are inverse to one another.
+Since F is finite by the definition of A1-category, it follows that E is finite.
+□
+Lemma 6.10. The set E is a neighborhood basis for an admissible topology on G, and E is
+a stabilizer class for the admissible group G.
+Proof. If α is the unique embedding of the trivial object into Ω then Gα = G; thus G belongs
+to E . It is clear that E is closed under conjugation. Lemma 6.8 shows that E is closed under
+finite intersections. It follows that E is a neighborhood basis for a topology on G, and the
+E is a stabilizer class.
+It remains to show that the topological group G is admissible. It is non-archimedean by
+construction. We now verify that it is Hausdorff. Thus suppose σ belongs to �
+U∈E U. Then
+for any embedding α: X → Ω we have σα = α. Since a map of ind-objects is determined by
+its restrictions to (non-ind) objects, it follows that σ is the identity, and so G is Hausdorff.
+Finally, we show G is Roelcke pre-compact. It suffices to show Gα0\G/Gβ0 is finite for two
+embeddings α0 : X → Ω and β0 : Y → Ω. This set is in bijection with G\(G/Gα0 × G/Gβ0).
+Since G acts transitively on Φ(X) with stabilizer Gα0, the set G/Gα (with its G-action) is
+identified with Φ(X); similarly, G/Gβ0 is identified with Φ(Y ). Thus finiteness follows from
+Lemma 6.9.
+□
+We have thus proved Theorem 6.5(a,b). Now, the action of G on Φ(X) is smooth, by
+definition of the topology on G. If α: X → Y is a morphism in A then there is an induced
+morphism α∗: Φ(Y ) → Φ(X) of G-sets. It follows that we have a functor
+Φ: A → T(G)op.
+To complete the proof of the theorem, we study properties of this functor in the next sequence
+of lemmas.
+Lemma 6.11. The functor Φ is faithful.
+Proof. Let α and β be two morphisms X → Y in C such that α∗ = β∗. Choose an embedding
+γ : Y → Ω, which is possible since Ω is universal. By assumption, we have γ ◦ α = γ ◦ β.
+Since γ is a monomorphism, it follows that α = β. Thus Φ is faithful.
+□
+
+22
+NATE HARMAN AND ANDREW SNOWDEN
+Lemma 6.12. The essential image of Φ is T(G; E ).
+Proof. For an object X of A, the G-set Φ(X) is isomorphic to G/Gα, where α ∈ Φ(X) is
+any element. We thus see that the essential image of Φ exactly consists of G-sets isomorphic
+to G/U with U ∈ E , which is exactly T(G; E ).
+□
+We have thus proved Theorem 6.5(c). We now turn our attention to Theorem 6.5(d). In
+what follows, we assume that A is an A-category.
+Lemma 6.13. The functor Φ is conservative; that is, if α: X → Y is a morphism in C such
+that α∗: Φ(Y ) → Φ(X) is an isomorphism then α is an isomorphism.
+Proof. Since A is an A-category, it is enough to show that α is an epimorphism.
+Thus
+suppose that β and γ are maps Y → Z such that β ◦ α = γ ◦ α. We thus have α∗β∗ = α∗γ∗.
+Since α∗ is an isomorphism, it follows that β∗ = γ∗. Since Φ is faithful, we find β = γ, as
+required.
+□
+Lemma 6.14. The functor Φ is full.
+Proof. Let X and Y be objects of C, and let ϕ: Φ(Y ) → Φ(X) be a map of G-sets. Choose an
+element β ∈ Φ(Y ), and let α = ϕ(β). Note that since ϕ is G-equivariant, we have Gβ ⊂ Gα.
+Let (Z, γ, δ) be an amalgamation of X and Y over 1, and let ǫ: Z → Ω be an embedding,
+as in Lemma 6.8. We have Gǫ = Gα ∩ Gβ = Gβ. Thus γ∗: Φ(Z) → Φ(Y ) is an isomorphism
+of G-sets; indeed, it is a G-equivariant map of transitive G-sets mapping ǫ to β, and ǫ and
+β have the same stabilizer in G. By the Lemma 6.13, it follows that γ is an isomorphism.
+Since the diagram in Lemma 6.8 is only defined up to isomorphism, we may as well suppose
+that Z = Y , γ = idY , and β = ǫ. We thus see that δ∗: Φ(Y ) → Φ(X) is a map of G-sets
+carrying β to α.
+Since Φ(Y ) is transitive, it follows that ϕ = δ∗, which completes the
+proof.
+□
+6.4. Fra¨ıss´e theory for B-categories. The following is our main theorem on B-categories,
+and contains Theorem 1.2 as a special case.
+Theorem 6.15. Let B be a B-category that is non-degenerate and has countably many
+isomorphism classes. Then there is a first-countable admissible group G and a stabilizer
+class E such that B is equivalent to S(G; E ). Moreover, if equivalence relations in B are
+effective (i.e., B is pre-Galois) then B is equivalent to S(G).
+Proof. Let A = A(B). By Theorem 5.8, this is an A-category satisfying the three conditions
+of Theorem 6.4.
+Thus by that theorem, we have A ∼= T(G; E ) for some first-countable
+admissible group G and stabilizer class E . We have equivalences B = B(A) and S(G; E ) =
+B(T(G; E )op), and so we obtain an equivalence B ∼= S(G; E ). The second statement follows
+from Proposition 4.9.
+□
+7. Examples from relational structures
+We now look at some examples of A-categories and B-categories coming from classes of
+relational structures. See §6.1 for basic definitions on relational structures.
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+23
+7.1. General comments. Let C be a non-empty class of finite relational structures. We
+assume throughout this section that C is hereditary and that |Cn| is finite for all n ≥ 0.
+Recall that we can regard C as a category, with morphisms being embeddings of structures.
+Proposition 7.1. The category C is an A1-category, and the following are equivalent:
+(a) C is an A-category.
+(b) Given Y ∈ C and a proper substructure X ⊂ Y , there exists a structure Z ∈ C and
+distinct embeddings Y ⇒ Z that have equal restriction to X.
+(c) Given Y ∈ C and a proper substructure X ⊂ Y , there exists a non-trivial self-
+amalgamation of Y over X.
+Proof. The class C contains the empty structure since it is non-empty and hereditary. It is
+clear that the empty structure is the initial object of C, and so C has an initial set. This
+verifies Definition 5.7(a).
+Let (β, γ) be a pre-amalgamation, where β : A → B and γ : A → C. Consider an object
+(δ, ǫ) of Amalg(β, γ), where δ: B → D and ǫ: C → D. We say that (δ, ǫ) is minimal if δ
+and ǫ are jointly surjective, i.e., D = im(δ) ∪ im(ǫ). Every object of Amalg(β, γ) admits a
+unique (up to isomorphism) map from a minimal object. Indeed, in the above notation, let
+D′ = im(δ) ∪ im(ǫ), regarded as a substructure of D. Then D′ is a minimal, with structure
+maps δ and ǫ, and the inclusion D′ → D is a map in Amalg(β, γ); a key point here is that
+D′ still belongs ot the class C since C is hereditary.
+Let S be a set of isomorphism class representatives for the minimal objects of Amalg(β, γ).
+The above argument shows that S is an amalgamation set for (β, γ). Since the cardinality of
+a minimal object is at most #B + #C and we have assumed |Cn| is finite for all n, it follows
+that S is finite. This verifies Definition 5.7(b).
+We have already explained (at the end of §5.4) how the remaining three conditions are
+equivalent.
+□
+Suppose that C is indeed an A-category and that it is also satisfies the amalgamation
+property; then C is a Fra¨ıss´e class. Let Ω be the Fra¨ıss´e limit, and let G = Aut(Ω), which
+acts oligomorphically on Ω. Theorem 6.4 gives an equivalence of A with T(G; E )op, where
+E is the set of subgroups of G of the form G(A) where A ⊂ Ω is a finite subset. (Recall that
+G(A) is the subgroup of G fixing each element of A.)
+7.2. Sets. Let C be the class of all finite sets (the signature in this case is empty). This is an
+A-category by Proposition 7.1. The amalgamation property holds. The Fra¨ıss´e limit is the
+countable set Ω = {1, 2, . . .} and its automorphism group is the infinite symmetric group S.
+Let E be the stabilizer class consisting conjugates of S(n), for variable n (see Example 2.1).
+Then we have an equivalence of categories C ∼= T(S; E )op.
+We can also describe the A-category T(S)op. Define a category C′ as follows. An object is
+a pair (X, G) where X is a finite set and G is a subgroup of the symmetric group Perm(X)
+on X. A morphism (X, G) → (Y, H) is an injective function α: X → Y such that H is
+contained in G, where here we identify Perm(X) with Perm(im(α)), which we in turn regard
+as a subgroup of Perm(Y ) in the usual manner. Then T(S)op is equivalent to C′.
+7.3. Total orders. Let C be the class of finite totally ordered sets (the signature consists of
+a single binary relation). This is an A-category by Proposition 7.1, and the amalgamation
+property holds. The Fra¨ıss´e limit Ω is the set of rational numbers, with its usual order.
+Let G = Aut(Ω). It turns out that every open subgroup of G has the form G(A) for some
+
+24
+NATE HARMAN AND ANDREW SNOWDEN
+finite subset A ⊂ Ω [HS1, Proposition 17.1]. We thus have an equivalence C ∼= T(G)op. The
+Delannoy category studied in [HSS] is associated to this group G.
+7.4. The countable matching. Let C be the class of all simple graphs in which each
+vertex belongs to at most one edge; the signature consists of a single binary relation (the
+edge relation on vertices). This is a Fra¨ıss´e class. The limit Ω is a perfect matching on a
+countable vertex set. Its automorphism group G is the wreath product Z/2 ≀ S, where S is
+the infinite symmetric group.
+The category C is an A1-category by Proposition 7.1, but it is not an A-category. To see
+this, let Y be a single edge, and let X ⊂ Y be one of the vertices. Then any map X → Z
+admits at most one extension to Y , and so Proposition 7.1(b) fails. Alternatively, the only
+self-amalgamation of Y over X is the trivial one, and so Proposition 7.1(c) fails.
+Theorem 6.5 does produce a faithful and essentially surjective functor Φ: C → T(G; E )op,
+for an appropriate stabilizer class E . We can see directly that this functor is not full: indeed,
+the map Φ(Y ) → Φ(X) is an isomorphism since every embedding X → Ω extends uniquely
+to Y . The inverse map does not come from a map Y → X in C, as there are no such maps.
+Let C0 be the (non-hereditary) subclass of C consisting of graphs in which each vertex
+belongs to exactly one edge. Then Φ restricts to an equivalence C0 → T(G; E )op.
+7.5. Permutation classes. Let P be the class of all finite sets equipped with a pair of total
+orders. Let X be a structure of P. Label the elements of X as 1, 2, . . . , n according to the
+first order. We can then enumerate the elements of X under the second order to get a string
+in the alphabet {1, . . . , n} in which each letter appears once. This string exactly determines
+the isomorphism type of X. We can thus view structures in P as permutations, and thus
+typically use symbols like σ for its members. The embedding order on P is the so-called
+containment order on partitions.
+A permutation class is a non-empty hereditary subclass C of P. There is an extensive
+literature on permutation classes; for an overview, see [Vat]. We mention one relevant result
+here: a theorem of Cameron [Cam2] asserts that there are exactly five permutation classes
+that are Fra¨ıss´e classes.
+Let σ be a permutation of length n, and let α1, . . . , αn be other permutations of lengths
+m1, . . . , mn. There is then a permutation σ[α1, . . . , αn] of length m = m1 + · · · + mn, called
+inflation. We refer to [Vat, §3.2] for the definition, and just give an example here:
+231[12, 321, 3412] = 56 987 3412.
+We have inserted spaces into the result to make the operation more clear. The three com-
+ponents on the right correspond to the three permutations in the brackets. Each uses an
+interval of numbers, and the order of the intervals is determined by the outside permuta-
+tion. A permutation class C is substitution closed if σ[α1, . . . , αn] belongs to C whenever
+σ, α1, . . . , αn all belong to C.
+Proposition 7.2. Let C be a substitution closed permutation class containing some permu-
+tation of length ≥ 2. Then C is an A-category.
+Proof. Let τ → σ be a non-isomorphism in C, and suppose the embedding misses i ∈ σ. Let
+n be the length of σ, and consider the inflation σ′ = σ[α1, . . . , αn] where αj = 1 for j ̸= i,
+and αi has length length 2. Note that C contains the permutation 1 and some permutation
+of length 2 since it is hereditary. One easily sees that σ′ is a non-trivial self-amalgamation
+of σ over τ. Thus C is an A-category by Proposition 7.1.
+□
+
+PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM
+25
+Example 7.3. A permutation is separable if it can be built from the permutation 1 with
+sums and skew-sums; the empty permutation is also separable. (Given two permutations
+α and β their sum is 12[α, β] and their skew-sum is 21[α, β].) Equivalently, a permutation
+is separable if the permutations 2413 and 3142 do not embed into it. The class C of all
+separable permutations is a substitution closed permutation class. It is thus an A-category
+by the above proposition.
+The class C does not have the amalgamation property. To see this, regard 123 as a subper-
+mutation of 1342 (using the first three positions) and 3124 (using the last three positions).
+In the class of all permutations, there is a unique amalgamation, namely 41352. This is not
+separable, since when the middle 3 is deleted we obtain 3142. However, C does have the
+joint embedding property, which means that any two objects embed into a common third
+object: indeed, if α and β are separable permutations then α and β each embed into their
+sum 12[α, β], which is also separable.
+Let B = B(C). Then B is a B-category. Since C has an initial object, the final object 1 of
+B is atomic. Since the amalgamation property fails for C, it follows that there are maps of
+atoms X → Z and Y → Z in B such that X ×Z Y = 0 (indeed, take X, Y , and Z to be the
+atoms corresponding to the permutations 1342, 3124, and 123 discussed above). However,
+since C has the joint embedding property, it follows that X × Y is non-empty for all atoms
+X and Y of B.
+□
+References
+[Cad]
+Anna Cadoret. “Galois categories” in Arithmetic and geometry around Galois theory. Progr. Math.,
+vol. 304, Birkh¨auser/Springer, Basel, 2013, pp. 171–246. DOI:10.1007/978-3-0348-0487-5 3
+[Cam1]
+Peter J. Cameron. Oligomorphic permutation groups. London Mathematical Society Lecture Note
+Series, vol. 152, Cambridge University Press, Cambridge, 1990.
+[Cam2]
+Peter J. Cameron. Homogeneous Permutations. Electron. J. Combin. 9 (2002), no. 3.
+DOI:10.37236/1674
+[Car]
+Olivia Caramello. Fraisse’s construction from a topos-theoretic perspective. Log. Univers. 8 (2014),
+no. 2, 261–281. DOI:10.1007/s11787-014-0104-6 arXiv:0805.2778
+[Del]
+P. Deligne. La cat´egorie des repr´esentations du groupe sym´etrique St, lorsque t n’est pas un entier
+naturel. In: Algebraic Groups and Homogeneous Spaces, in: Tata Inst. Fund. Res. Stud. Math.,
+Tata Inst. Fund. Res., Mumbai, 2007, pp. 209–273.
+Available at: https://www.math.ias.edu/files/deligne/Symetrique.pdf
+[DG1]
+Manfred Droste, R¨udier G¨obel. A categorial theorem on universal objects and its application
+in abelian group theory and computer science. Contemp. Math. 131 (Part 3) 1992, pp. 49–74.
+DOI:10.1090/conm/131.3
+[DG2]
+Manfred Droste, R¨udier G¨obel. Universal domains and the amalgamation property. Math. Struc-
+tures Comput. Sci. 3 (1993), no. 2, pp. 137–159. DOI:10.1017/S0960129500000177
+[DM]
+P. Deligne, J. Milne. Tannakian Categories. In “Hodge cycles, motives, and Shimura varieties,”
+Lecture Notes in Math., vol. 900, Springer–Verlag, 1982. DOI:10.1007/978-3-540-38955-2 4
+Available at: http://www.jmilne.org/math/xnotes/tc.html
+[Fra]
+R. Fra¨ıss´e. Sur certaines relations qui g´en´eralisent l’order des nombres rationnels. C. R. Acad. Sci.
+237 (1953), pp. 540–542.
+[GL]
+Wee Liang Gan, Liping Li. Noetherian property of infinite EI categories. New York J. Math. 21
+(2015) pp. 369–382. arXiv:1407.8235
+[HS1]
+Nate Harman, Andrew Snowden. Oligomorphic groups and tensor categories. arXiv:2204.04526
+[HS2]
+Nate Harman, Andrew Snowden. Ultrahomogeneous tensor structures. arXiv:2207.09626
+[HS3]
+Nate Harman, Andrew Snowden. Oligomorphic component groups of pre-Tannakian categories. In
+preparation.
+[HSS]
+Nate Harman, Andrew Snowden, Noah Snyder. The Delannoy category. arXiv:2211.15392
+
+26
+NATE HARMAN AND ANDREW SNOWDEN
+[Irw]
+Trevor L. Irwin. Fra¨ıss´e limits and colimits with applications to continua. Ph. D. Thesis, Indiana
+University, 2007.
+[Kub]
+Wies�law Kubi´s. Fra¨ıss´e sequences: category-theoretic approach to universal homogeneous struc-
+tures. arXiv:0711.1683
+[Mac]
+Dugald Macpherson. A survey of homogeneous structures. Discrete Math. 311 (2011), no. 15,
+pp. 1599–1634. DOI:10.1016/j.disc.2011.01.024
+[Ost]
+Victor Ostrik. On symmetric fusion categories in positive characteristic. Selecta Math. N.S. 26
+(2020). DOI:10.1007/s00029-020-00567-5 arXiv:1503.01492
+[Stacks] Stacks Project. http://stacks.math.columbia.edu (accessed 2022).
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+B´ona. CRC Press, 2015. arXiv:1409.5159
+

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+arXiv:2301.00279v1  [math.AC]  31 Dec 2022
+A NOTE ON WEAK w-PROJECTIVE MODULES
+REFAT ABDELMAWLA KHALED ASSAAD
+Abstract. Let R be a ring. An R-module M is said to be a weak w-projective
+module if Ext1
+R(M, N) = 0 for all N ∈ P†∞
+w
+(see, [18]). In this paper, we in-
+troduce and study some properties of weak w-projective modules. And we use
+these modules to characterize some classical rings, for example, we will prove
+that a ring R is a DW -ring if and only if every weak w-projective is projective,
+R is a Von Neumann regular ring if and only if every FP-projective is weak w-
+projective if and only if every finitely presented R-module is weak w-projective
+and R is a w-semi-hereditary if and only if every finite type submodule of a
+free module is weak w-projective if and only if every finitely generated ideal
+of R is a weak w-projective.
+1. Introduction
+In this paper, all rings are considered commutative with unity and all modules
+are unital. Let R be a ring and M be an R-module. As usual, we use pdR(M),
+idR(M) and fdR(M) to denote, respectively, the classical projective dimension,
+injective dimension and flat dimension of M, and w.gl.dim(R) and gl.dim(R) to
+denote, respectively, the weak and global homological dimensions of R.
+Now, we review some definitions and notation. Let J be an ideal of R. Following
+[23], J is called a Glaz-Vasconcelos ideal (a GV -ideal for short) if J is finitely gener-
+ated and the natural homomorphism ϕ : R → J∗ = HomR(J, R) is an isomorphism.
+Note that the set GV (R) of GV -ideals of R is a multiplicative system of ideals of
+R. Let M be an R-module. It is Defined
+torGV (M) = {x ∈ M | Jx = 0 for some J ∈ GV (R)}.
+It is clear that torGV (M) is submodule of M. M is said to be GV -torsion (resp.,
+GV -torsion-free) if torGV (M) = M (resp., torGV (M) = 0).
+A GV -torsion-free
+module M is called a w-module if Ext1
+R(R/J, M) = 0 for any J ∈ GV (R). Then,
+projective modules and reflexive modules are w-modules. In the recent paper [23],
+it was shown that flat modules are w-modules. Also it is known that a GV -torsion-
+free R-module M is a w-module if and only ExtR
+1 (N, M) = 0 for every GV -torsion
+R-module N (see, [13], Theorem 6.2.7). The notion of w-modules was introduced
+firstly over a domain [17] in the study of Strong Mori domains and was extended
+to commutative rings with zero divisors in [23]. Let w − Max(R) denote the set
+of w-ideals of R maximal among proper integral w-ideals of R (maximal w-ideals).
+Following [23, Proposition 3.8], every maximal w-ideal is prime. For any GV -torsion
+free module M,
+Mw := {x ∈ E(M) | Jx ⊆ M for some J ∈ GV (R)}
+2010 Mathematics Subject Classification. 13D05, 13D07, 13H05.
+Key words and phrases. projective modules , weak w-projective modules, w-flat, GV -torsion,
+finitely presented type, DW -rings, coherent rings, w-coherent rings.
+1
+
+2
+R.A.K. ASSAAD
+is a w-submodule of E(M) containing M and is called the w-envelope of M, where
+E(M) denotes the injective hull of M. It is clear that a GV -torsion-free module M
+is a w-module if and only if Mw = M.
+Let M and N be R-modules and let f : M → N be a homomorphism. Following
+[12], f is called a w-monomorphism (resp., w-epimorphism, w-isomorphism) if fm :
+Mm → Nm is a monomorphism (resp., an epimorphism, an isomorphism) for all
+m ∈ w − Max(R). A sequence A → B → C of modules and homomorphisms is
+called w-exact if the sequence Am → Bm → Cm is exact for all m ∈ w − Max(R).
+An R-module M is said to be of finite type if there exists a finitely generated free
+R-module F and a w-epimorphism g : F → M. Similarly, an R-module M is said to
+be of finitely presented type if there exists a w-exact sequence F1 → F0 → M → 0,
+where F1 and F0 are finitely generated free.
+In recent years, homological theoretic characterization of w-modules has received
+attention in several papers the literature (for example see [[1], [21], [19], [18]]).
+The notion of w-projective modules and w-flat modules appeared first in [11] when
+R is an integral domain and was extended to an arbitrary commutative ring in
+[[14], [2]].
+In [14], F. G. Wang and H. Kim generalized projective modules to
+w-projective modules by the w-operation.
+An R-module M is said to be a w-
+projevtive if Ext1
+R(L(M), N) is GV -torsion for any torsion-free w-module N, where
+L(M) = (M/ torGV (M))w. Denote by Pw the class of all w-projective R-modules.
+Following [20], an R-module M is a w-split if and only if Ext1
+R(M, N) is GV -torsion
+for all R-modules N. Denote by Sw the class of all w-split R-modules. Hence, by
+[[?], Corollary 2.4], every w-split module is w-projective.
+Following [2], an R-
+module M is said to be w-flat if for any w-monomorphism f : A → B, the induced
+sequence 1 ⊗ f : M ⊗R A → M ⊗R B is a w-monomorphism. Denote by Fw the
+class of all w-flat R-modules. Following [18], throughout this paper, P†∞
+w
+denote
+the class of GV -torsion-free R-modules N with the property that Extk
+R(M, N) = 0
+for all w-projective R-modules M and for all integers k ≥ 1 Clearly, every GV -
+torsionfree injective R-module belongs to P†∞
+w .
+An R-module M is said to be
+weak w-projective if Ext1
+R(M, N) = 0 for all N ∈ P†∞
+w : Denote by wPw the class
+of all weak w-projective modules. Following [18], Wang and Qiao introduce the
+notions of the weak w-projective dimension (w.w-pd) of a module and the global
+weak w-projective dimension (gl.w.w-dim) of a ring. Following [18], a GV -torsion-
+free module M is said to be a strong w-module if Exti
+R(N, M) = 0for any integer
+i ≥ 1 and all GV -torsion modules N. Denote by W∞ the class of all strong w-
+modules. Then all GV -torsion-free injective modules are strong w-modules. Clearly,
+P†∞
+w
+⊆ W∞. But, in [18], they do not showed that P†∞
+w
+and W∞ are the different
+class of R-modules, and this question was answered in [8].
+Recall from [4] that an R-module M is called FP-projective if Ext1
+R(M, N) = 0
+for any absolutely pure R-module N. Denote by FP the class of all FP-projective
+modules. Recall that an R-module A is an absolutely pure if A is a pure submodule
+in every R-module which contains A as a submodule (see, [3]). C. Megibbeni showed
+in [5], that an R-module A is absolutely pure if and only if Ext1
+R(F, A) = 0, for
+every finitely presented module F. Hence, an absolutely pure module is precisely
+a FP-injective module in [7].
+
+A NOTE ON WEAK w-PROJECTIVE MODULES
+3
+2. results
+In this section, we introduce a characterize of some classical ring. But we need
+the following lemma
+Lemma 2.1. ([18], Proposition 2.5) An R-module M is weak w-projective if Ext1
+R(M, N) =
+0 for all N ∈ P†∞
+w
+and for all k ≥ 0.
+It is obvious that, for the class of modules
+{ projective } ⊆ { w-split } ⊆ { w-proective } ⊆ {weak w-proective } ⊆ { w-flat }.
+By [[1], Proposition 2.5], if R is a perfect ring, then the five classes of modules above
+coincide.
+In the following proposition, we will give some characterizations of weak w-
+projective modules.
+Proposition 2.2. Let M be an R-module. Then the following are equivalent:
+(1) M is weak w-projective.
+(2) M ⊗ F is weak w-projective for any projective R-module F.
+(3) HomR(F, M) is weak w-projective for any finitely generated projective R-
+module F.
+(4) For any exact sequence of R-modules
+0 → A → B → C → 0
+with A ∈
+P†∞
+w , the sequence 0 → HomR(M, A) → HomR(M, B) → HomR(M, C) → 0
+is exact.
+(5) For any w-exact sequence of R-modules 0 → L → E → M → 0
+the se-
+quence 0 → HomR(M, N) → HomR(E, N) → HomR(L, N) → 0
+is exact
+for any R-module N ∈ P†∞
+w .
+(6) For any exact sequence of R-modules 0 → L → E → M → 0 the sequence
+0 → HomR(M, N) → HomR(E, N) → HomR(L, N) → 0
+is exact for any
+R-module N ∈ P†∞
+w .
+Proof. (1) ⇒ (2). Let F be a projective R-module. For any R-module N in P†∞
+w ,
+we have Ext1
+R(F ⊗M, N) ∼= HomR(F, Ext1
+R(M, N)) by [[13], Theorem 3.3.10]. Since
+M is a weak w-projective, Ext1
+R(M, N) = 0. Thus, Ext1
+R(F ⊗ M, N) = 0. Hence,
+F ⊗ M is a weak w-projective.
+(2) ⇒ (1) and (3) ⇒ (1). Follow by letting F = R.
+(1) ⇒ (3). Let N ∈ P†∞
+w , for any finitely generated projective R-module F, we
+have F ⊗ Ext1
+R(M, N) ∼= Ext1
+R(HomR(F, M), N) by [[13], Theorem 3.3.12]. Since
+M is weak w-projective, so Ext1
+R(M, N) = 0. Hence, Ext1
+R(HomR(F, M), N) = 0,
+which implies that HomR(F, M) is weak w-projective.
+(1) ⇒ (4). Let 0 → A → B → C → 0 be an exact sequence with A ∈ P†∞
+w , then
+we have the exact sequence 0 → HomR(M, A) → HomR(M, B) → HomR(M, C) →
+Ext1
+R(M, A). Since M is weak w-projective and A ∈ P†∞
+w , so Ext1
+R(M, A) = 0.
+Thus, 0 → HomR(M, A) → HomR(M, B) → HomR(M, C) → 0 is exact.
+(4) ⇒ (1). Let N ∈ P†∞
+w , consdier an exact sequence 0 → N → E → L → 0
+with E is injective module, then we have the exact sequence 0 → HomR(M, N) →
+HomR(M, E) → HomR(M, L) → Ext1
+R(M, N) → 0, and keeping in mind that
+0 → HomR(M, N) → HomR(M, E) → HomR(M, L) → 0 is exact, we deduce that
+Ext1
+R(M, N) = 0. Hence, M is weak w-projective.
+(1) ⇒ (5).
+Let 0 → L → E → M → 0
+be a w-exact sequence. For any R-
+module N ∈ P†∞
+w
+so N ∈ W∞. By [[18], Lemma 2.1], we have the exact sequence
+
+4
+R.A.K. ASSAAD
+0 → HomR(M, N) → HomR(E, N) → HomR(L, N) → Ext1
+R(M, N). Since M is
+weak w-projective, so Ext1
+R(M, N) = 0, and (5) is holds.
+(5) ⇒ (6). Trivial.
+(6) ⇒ (1). Let 0 → L → E → M → 0 be an exact sequence with E is projective.
+Hence, for any R-module N ∈ P†∞
+w , we have 0 → HomR(M, N) → HomR(E, N) →
+HomR(L, N) → Ext1
+R(M, N) → 0 is exact sequence, and keeping in mind that
+0 → HomR(M, N) → HomR(E, N) → HomR(L, N) → 0 is exact, we deduce that
+Ext1
+R(M, N) = 0, which implies that M is weak w-projective.
+□
+Recall from [12], that a ring is said to be w-coherent if every finitely generated
+ideal of R is of finitely presented type.
+Proposition 2.3. Let R be a w-coherent ring, E be an injective R-module, M be a
+finitely presented type and N be an R{x}-module. Then, if M is weak w-projective
+R-module, so TorR
+n (M, Hom(N, E)) = 0.
+Proof. Let M be a weak w-projective R-module and let N be an R{x}-modul, so
+Extn
+R(M, n) = 0 by [[18], Proposition 2.5] and since every R{x}-module in P†∞
+w
+by
+[[18], Proposition 2.4]. Henace, by [[16], Proposition 2.13(6)], we have
+TorR
+n (M, Hom(N, E)) ∼= Hom(Extn
+R(M, N), E) = 0.
+Which implies that, TorR
+n (M, Hom(N, E)) = 0
+□
+Proposition 2.4. Every weak w-projective of finite type is of finitely presented
+type.
+Proof. Let M be a weak w-projective R-module of finite type, so by [[18], Corollary
+2.9] M is w-projective of finite type. Thus, by [[13], Theorem 6.7.22], we have M
+is finitely presented type.
+□
+Proposition 2.5. Let M be a GV -torsion-free module. The following assertions
+hold.
+(1) Mw/M is a weak w-projective module.
+(2) M is a weak w-projective if and only if so is Mw.
+Proof. (1). Let M be a GV -torsion-free module. So, by [[13], Proposition 6.2.5] we
+have Mw/M a GV -torsion module. Hence, by [[18], Proposition 2.3(2)], we have
+Mw/M is weak w-projective.
+(2). Let N be an R-module in P†∞
+w . Since M is GV -torsion-free, we have by (1)
+Mw/M is weak w-projective module. Consider the following exact sequence
+0 → M → Mw → Mw/M → 0
+which is w-exact. Hence, by [[18], Proposition 2.5], M is weak w-projective if and
+only if Mw is weak w-projective.
+□
+Recall that a ring R is called a DW-ring if every ideal of R is a w-ideal, or
+equivalently every maximal ideal of R is w-ideal [6]. Examples of DW-rings are
+Pr¨ufer domains, domains with Krull dimension one, and rings with Krull dimension
+zero. We note that if R is DW-ring, then every R-module in P†∞
+w .
+In the following proposition, we will give a new characterizations of DW-rings which
+are the only rings with these properties.
+Proposition 2.6. Let R be a ring. The following statements are equivalent:
+
+A NOTE ON WEAK w-PROJECTIVE MODULES
+5
+(1) Every weak w-projective R-module is projective.
+(2) Every w-projective R-module is projective.
+(3) Every GV -torsion R-module is projective.
+(4) Every GV -torsion-free R-module is strong w-module.
+(5) Every finitely presented type w-flat is projective.
+(6) Every weak w-projective R-module is w-module.
+(7) R is DW-ring.
+Proof. (1) ⇒ (2) and (2) ⇒ (3). The are trivial.
+(3) ⇒ (4). Let M be a GV -torsion-free R-module, for any GV -torsion R-module
+N, we have Exti
+R(N, M) = 0 since N is projective. Hence, M is a strong w-module.
+(4) ⇒ (7). By [[12], Theorem 3.8] since every strong w-module is w-module.
+(1) ⇒ (6). Trivial, since every projective R-module is w-module.
+(2) ⇒ (5). Let M be a finitely presented type w-flat. By [[18], Corollary 2.9], we
+have M is a finite type w-projective. Hence, M is a projective R-module by (2).
+(5) ⇒ (7). Let M be a finitely presented w-flat. Then, M is finitely presented type
+w-flat, so M is projective by (5). Hence, by [[9], Proposition 2.1], we have R is a
+DW-ring.
+(6) ⇒ (7). Let M be a GV -torsion-free R-module. Hence, by Proposition 2.5,
+we have Mw/M is weak w-projectiveis and so w-module by (6). Thus, Mw/M is
+a GV -torsion-free. Hence, Mw/M = 0 and so Mw = M. Thus, M is w-module.
+Then, R is a DW-ring by [[12], Theorem 3.8].
+(7) ⇒ (1).
+Let M be a weak w-projective.
+For any R-module N, we have
+Ext1
+R(M, N) = 0 because N ∈ P†∞
+w
+(since R is DW). Hence, M is a projective
+module.
+□
+Note that the equivalence (1) ⇔ (7) in Proposition 2.6 was given in [[8], Propo-
+sition 4.4] for the domain case.
+L. Mao and N. Ding in [[4]], proved that a ring R is a Von Neumann regular if
+and only if every FP-projective R-module is projective.
+Next, we will give new characterizations of a Von Neumann regular rings by weak
+w-projective modules.
+Proposition 2.7. Let R be a ring. Then, the following statements are equivalent:
+(1) Every FP-projective R-module is weak w-projective.
+(2) Every finitely presented R-module is weak w-projectiv.
+(3) Every finitely presented R-module is w-flat.
+(4) R is a Von Neumann regular.
+Proof. (1) ⇒ (2). Follows from the fact that every finitely presented R-module is
+FP-projective.
+(2) ⇒ (3). Let M be a finitely presented R-module, so M is weak w-projective.
+Hence, M is w-flat by [[18], Corollary 2.11].
+(3) ⇒ (4). Let I be a finitely generated ideal of R, then R/I is finitely presented.
+So R/I is w-flat by (3), then w − fdR(R/I) = 0. Thus, w − w.gl.dim(R) = 0 by
+[[19], Proposition 3.3]. Hence, R is Von Neumann regular by [[15], Theorem 4.4].
+(4) ⇒ (1). Let M be a FP-projective, so M is projective by [[4], Remarks 2.2].
+Hence, M is a weak w-projective.
+□
+Next, we will give an example of FP-projective module which is not weak w-
+projective.
+
+6
+R.A.K. ASSAAD
+Example 2.8. Consider the local Quasi-Frobenius ring R := k[X]/(X2) where k
+is a field, and denote by X the residue class in R of X. Then, (X) is FP-projective
+R-module which is not weak w-projective.
+Proof. Since R is a Quasi-Frobenius ring, then every absolutely pure R-module is
+injective. Hence, for any absolutely pure R-module N, we have Ext1
+R((X), N) = 0,
+so (X) is FP-projective. But, (X) is not projective by [[10], Example 2.2], and so
+not weak w-projective, since R is DW-ring.
+□
+Recall from [[15]] that a ring R is said to be w-semi-hereditary if every finite
+type ideal of R is w-projective.
+Proposition 2.9. The following are equivalent:
+(1) R w-semi-hereditary.
+(2) Every finite type submodule of a free module is weak w-projective.
+(3) Every finite type ideal of R is a weak w-projective.
+(4) Every finitely generated submodule of a free module is weak w-projective.
+(5) Every finitely generated ideal of R is a weak w-projective
+Proof. (1) ⇒ (2). Let J be a finite type submodule of a any free module. Hence,
+J is w-projective by [[15], Theorem 4.11]. Then J is weak w-projective by [[18],
+Corollary 2.9].
+(2) ⇒ (3) ⇒ (5) and (2) ⇒ (4) ⇒ (5). These are trivial.
+(5) ⇒ (1). Let J be a finite type ideal of R. Then J is w-isomorphic to a finitely
+generated subideal I of J. Hence J is weak w-projective by hypothesis and [[18],
+Corollary 2.7].
+□
+Proposition 2.10. Every GV -torsion-free weak w-projective module is torsion-
+free.
+Proof. Let M be a GV -torsion-free weak w-projective module. Hence, M is a GV -
+torsion-free w-flat by [[18], Corollary 2.11].Thus, by [[13], Proposition 6.7.6], we
+have M is torsion-free.
+□
+In the next example we will prove that a weak w-projective module need not to
+be torsion-free.
+Example 2.11. Let R be an integral domain and J be a proper GV -ideal of R.
+Then M := R ⊕ R/J is a weak w-projective module but not torsion-free.
+Proposition 2.12. Let R be a ring and M be a finitely presented R-module. Then,
+the following statements are equivalent:
+(1) M is w-split.
+(2) M is weak w-projective.
+(3) For any w-exact 0 → A → B → C → 0 , the sequence
+0 → HomR(M, A) → HomR(M, B) → HomR(M, C) → 0 is w-exact.
+Proof. (1) ⇒ (2). Trivial, since every w-split R-module is weak w-projective.
+(2) ⇒ (3). Let 0 → A → B → C → 0 be a w-exact sequence of R-modules. Then,
+for any maximal w-ideal m of R, 0 → Am → Bm → Cm → 0 is exact sequence of
+Rm-modules. Thus, since Mm is free by [[18], Proposition 2.8], we have the exact
+
+A NOTE ON WEAK w-PROJECTIVE MODULES
+7
+sequence 0 → HomR(Mm, Am) → HomR(Mm, Bm) → HomR(Mm, Cm) → 0 . Since
+M is finitely presented, we have the commutative diagram
+HomRm(Mm, Am)
+→
+HomRm(Mm, Bm)
+→
+HomRm(Mm, Cm)
+|| ≀
+|| ≀
+|| ≀
+HomR(M, A)m
+→
+HomR(M, B)m
+→
+HomR(M, C)m
+Thus, 0 → HomR(M, A)m → HomR(M, B)m → HomR(M, C)m → 0 is exact, and
+so, 0 → HomR(M, A) → HomR(M, B) → HomR(M, C) → 0 is w-exact.
+(3) ⇒ (1). By [[20], Proposition 2.4].
+□
+Recall from [22], that a w-exact sequence of R-modules 0 → A → B → C → 0
+is said to be w-pure exact if, for any R-module M, the induced sequence
+0 → A ⊗ M → B ⊗ M → C ⊗ M → 0
+is w-exact.
+Proposition 2.13. Let C be a finitely presented type R-module. Then, the follow-
+ing statements are equivalent:
+(1) C is a weak w-projective R-module.
+(2) Every w-exact sequence of R-modules 0 → A → B → C → 0 is w-pure
+exact.
+Proof. (1) ⇒ (2). Since every weak w-projective is w-flat by [[18], Corollary 2.11].
+Hence, by [[22], Theorem 2.6], we have the result.
+(2) ⇒ (1). Let 0 → A → B → C → 0 be a w-exact sequence, so is a w-pure
+exact by hypothesis. Thus, C is w-flat by [[22], Theorem 2.6]. Hence C is a weak
+w-projective by [[18], Corollary 2.9].
+□
+Proposition 2.14. The following are equivalent for a finite type R-module M.
+(1) M is a w-projective module.
+(2) Ext1
+R(M, B) = 0 for any B ∈ P†∞
+w .
+(3) Ext1
+R(M, N) = 0 for any R{x}-module N.
+(4) M{x} is a projective R{x}-module.
+Proof. (1) ⇒ (2). This is trivial.
+(2) ⇒ (3). By [[18], Proposition 2.4].
+(3) ⇒ (4). Let N be an R{x}-module, we have by [[16], Proposition 2.5],
+Extn
+R{x}(M{x}, N) ∼= Extn
+R(M, N) = 0.
+Thus, M{x} is a projective R{x}-module.
+(4) ⇒ (1). Let M{x} be a projective R{x}-module, so M{x} is finitely generated
+by [[13], Theorem 6.6.24] and since M is of finite type. Hence, by [[13], Theorem
+6.7.18], M is w-projective module.
+□
+Recall form [[18]], that an R-module D is said to be P†∞
+w -divisible if it is iso-
+morphic to E/N where E is a GV -torsin-free injective R-module and N ∈ P†∞
+w
+is
+a submodule of E.
+Proposition 2.15. Let M be an R-module and any integer m ≥ 1. The following
+are equivalent.
+(1) w.w-pdRM ≤ m.
+(2) Extm
+R (M, D) = 0 for all P†∞
+w -divisible R-module D.
+
+8
+R.A.K. ASSAAD
+Proof. (1) ⇒ (2). Let N ∈ P†∞
+w . Then there exists an exact sequence of R-modules
+0 → N → E → H → 0, where E is a GV -torsion-free injective R-module. Hence,
+D is P†∞
+w -divisibl. Then we have the induced exact sequence
+Extm
+R (M, H) → Extm+1
+R
+(M, N) → Extm+1
+R
+(M, E) = 0,
+for any integer m ≥ 1. The left term is zero by hypothesis. Hence, Extm+1
+R
+(M, N) =
+0, which implies that w.w-pdRM ≤ m by [[18], Proposition 3.1].
+(2) ⇒ (1). Let w.w-pdRM ≤ m and D be a P†∞
+w -divisible R-module. Then we
+have an exact sequence 0 → N → E → H → 0, where E is a GV -torsion-free
+injective R-module and N ∈ P†∞
+w . Hence, we have the exact sequence
+0 = Extm
+R (M, E) → Extm
+R (M, H) → Extm+1
+R
+(M, N).
+The right term is zero by [[18], Proposition 3.1]. Therefore, Extm
+R (M, H) = 0.
+□
+Proposition 2.16. Let M and N be two R-modules. Then,
+w.w-pdR(M ⊕ N) = sup{w.w-pdRM, w.w-pdRN}
+Proof. The inequality w.w-pdR(M ⊕ N) ≤ sup{w.w-pdRM, w.w-pdRN} follows
+from the fact that the class of weak w-projective modules is closed under direct
+sums by [[18], Proposition 2.5(1)]. For the converse inequality, we may assume that
+w.w-pdR(M ⊕ N) = n is finite. Thus, for any R-module X ∈ P†∞
+w ,
+Extn+1
+R
+(M ⊕ N, X) ∼= Extn+1
+R
+(M, X) ⊕ Extn+1
+R
+(N, X).
+Since Extn+1
+R
+(M ⊕ N, X) = 0 by [[18], Proposition 3.1]. Hence, Extn+1
+R
+(M, X) =
+Extn+1
+R
+(N, X) = 0, which implies that, sup{w.w-pdRM, w.w-pdRN} ≤ n.
+□
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+J. of Algebra. 5 (2011) 1219 -1224. 6
+[11] F. Wang, On w-projective modules and w-flat modules, Algebra Colloq. 4 (1997), no. 1,
+111-120. 2
+[12] F. Wang, Finitely presented type modules and w-coherent rings, J. Sichuan Normal Univ.
+33 (2010) 1–9. 2, 4, 5
+[13] F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, (Springer
+Nature Singapore Pte Ltd., Singapore, 2016). 1, 3, 4, 6, 7
+[14] F. Wang and H. Kim, Two generalizations of projective modules and their applications, J.
+Pure Applied Algebra 219 (2015) 2099-2123. 2
+
+A NOTE ON WEAK w-PROJECTIVE MODULES
+9
+[15] F. Wang and H. Kim, w-injective modules and w-semi-hereditary rings, J. Korean Math.
+Soc. 51 (2014), no. 3, 509–525. 5, 6
+[16] F. Wang and H. Kim, Relative FP-injective modules and relative IF rings, Commun. Alge-
+bra, Vol. 49, (2021), 3552-3582. 4, 7
+[17] F. Wang and R. L. McCasland, On w-modules over strong Mori domains, Comm. Algebra
+25(4), 1285-1306 (1997). 1
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+(2019). 1, 2, 3, 4, 5, 6, 7, 8
+[19] F. Wang and L. Qiao, The w-weak global dimension of commutative rings, Bull. Korean
+Math. Soc. 52 (2015), no. 4, 1327–1338. 2, 5
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+Korean. Math. Soc. 48(1) (2011) 207–222.
+1
+Department of Mathematics, Faculty of Science, University Moulay Ismail Meknes,
+Box 11201, Zitoune, Morocco
+Email address: [email protected]
+

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+arXiv:2301.04654v1  [nlin.CD]  11 Jan 2023
+Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short
+Correlations in Singlets and Doublets
+T. Ara´ujo Lima1, ∗ and R. B. do Carmo2, †
+1Departamento de F´ısica, Universidade Federal Rural de Pernambuco, Recife, PE 52171-900, Brazil
+2Instituto Federal de Alagoas, Piranhas, AL 57460-000, Brazil
+(Dated: January 13, 2023)
+Billiards are flat cavities where a particle is free to move between elastic collisions with the bound-
+ary. In chaos theory these systems are simple prototypes, their conservative dynamics of a billiard
+may vary from regular to chaotic, depending only on the border. The results reported here seek
+to shed light on the quantization of classically chaotic systems. We present numerical results on
+classical and quantum properties in two bi-parametric families of Billiards, Elliptical Stadium Bil-
+liard (ESB) and Elliptical-C3 Billiards (E-C3B). Both are elliptical perturbations of chaotic billiards
+with originally circular sectors on their borders. Our numerical calculations show evidence that the
+elliptical families can present a mixed classical phase space, identified by a parameter ρc < 1,
+which we use to guide our analysis of quantum spectra. We explored the short correlations through
+nearest neighbor spacing distribution p(s), which showed that in the mixed region of the classical
+phase space, p(s) is well described by the Berry-Robnik-Brody (BRB) distributions for the ESB.
+In agreement with the expected from the so-called ergodic parameter α = tH/tT, the ratio between
+the Heisenberg time and the classical diffusive-like transport time signals the possibility of quantum
+dynamical localization when α < 1. For the E-C3B family, the eigenstates can be split into singlets
+and doublets. BRB describes p(s) for singlets as the previous family in the mixed region. However,
+the p(s) for doublets are described by new distributions recently introduced in the literature but
+only tested in a few cases for ρc < 1. We observed that as ρc decreases, the p(s)’s tend to move
+away simultaneously from the GOE (singlets) and GUE (doublets) distributions.
+Keywords: Billiards. Chaos. Quantization. GOE. GUE.
+I.
+INTRODUCTION
+The idea that molecules may be behind Thermody-
+namics (grounded in Statistical Mechanics) was one of
+the tremendous scientific advances of the 19th century. In
+particular, these particles, constituents of gases, are asso-
+ciated with the concept of ergodicity, then called molecu-
+lar chaos. The word ergodic came from the Greek ergon
+(work) and odos (trajectory) and was used by Boltzmann
+to represent the hypothetical visit to all points of the
+phase space by a particle of that gas with random micro-
+scopic dynamic behavior. The introduction of the proba-
+bility in theory that came to be called Statistical Mechan-
+ics of Equilibrium passed by a long probationary regime,
+with more convincing results occurring only in the first
+decades of the 20th century [1]. The so-called Ergodic
+Hypothesis only gained the rigor of a theorem with the
+work of the Russian mathematician Y. Sinai in the 60s-
+70s for an ideal gas of only two particles [2]. A system is
+chaotic if two neighboring trajectories in the phase space
+separate exponentially.
+Suppose the distance in phase
+space between such trajectories is proportional to eλt.
+The λ parameter is called the Lyapunov exponent. In re-
+ality, λ represents the greatest of Lyapunov’s exponents.
+Therefore, the existence of at least one positive Lyapunov
+exponent characterizes a chaotic system [3]. Billiards sys-
+tems are prototypes in the study of chaos and describe
+∗ Corresponding author: [email protected]
+† [email protected]
+the free movement of a point particle in a closed domain
+Ω with elastic reflections on the boundary ∂Ω of the do-
+main. The nature of this conservative dynamical system
+depends exclusively on the shape of the border ∂Ω, vary-
+ing from entirely regular (i.e., ellipses and annular con-
+centric regions) to completely chaotic (i.e., Sinai billiard).
+Without loss of generality, we consider that the particle
+has mass m = 1 and velocity of module |v| = 1. A dis-
+crete dynamics well describes this 2-dimensional motion
+in time on variables (ℓ, φ), the fraction of perimeter of
+∂Ω, and the incidence angle where a collision happens
+parametrizes the discrete-time generally [4]. A primor-
+dial example that deserves to be mentioned here is the
+Bunimovich stadium. This billiard can present λ > 0.
+Its shape consists of two semicircles joined by two finite-
+size segments 2t, forming a stadium.
+It is chaotic for
+any t > 0. In a pure circular billiard, collisions keep the
+angular momentum in relation to its center (focus) con-
+stant.
+The Bunimovich stadium does not present this
+property in its dynamics, known as a defocusing [5]. It is
+hugely relevant to this work because the Elliptical Sta-
+dium Billiard is a perturbation of it, resulting in classical
+dynamics with mixed phase space.
+Quantum mechanics has been one of the best-tested
+physical theories since its emergence. The theory makes
+excellent predictions not only for the atom of hydro-
+gen, which is classically integrable, as well as the he-
+lium atom, which is classically not integrable. Nothing is
+more natural than whether there is an effect analogous to
+chaos in quantum mechanics. The term quantum chaos
+is generally understood as studying the quantum behav-
+
+2
+ior of classically chaotic systems [6]. One commonly used
+means of studying these systems is to statistically char-
+acterize spectral properties in the semiclassical regime
+and compare them with results from the random matri-
+ces theory [7].
+In billiards, obtaining the energy spectrum is an essen-
+tial step for analysis. The problem is to solve the time-
+independent Schr¨odinger equation with null potential in
+the planar region Ω with Dirichlet boundary conditions
+at ∂Ω:
+�
+∇2ϕn(r) = −k2
+nϕ(r),
+r ∈ Ω
+ϕn(r) = 0,
+r ∈ ∂Ω,
+(1)
+expression is also known as the Helmholtz Equation [8].
+Where k2
+n = 2mEn/ℏ2. In order to characterize univer-
+sality, one must first unfold the energy spectrum {En}
+so that a unit means (⟨sn⟩ = 1) nearest neighbor spacing
+(nns) sn = En+1 − En is obtained. This approach be-
+came relevant after two important conjectures. Namely,
+the Berry-Tabor (BT) conjecture [9] and the Bohigas-
+Giannoni-Schmit (BGS) conjecture [10].
+The BT con-
+jecture states that, in the semiclassical limit, the statis-
+tical properties of the energy spectrum of a classically
+integrable system must correspond to the prediction of
+uncorrelated randomly distributed energy levels.
+As a
+result, the semiclassical nns distribution p(s) must obey
+Poisson:
+pP(s) = exp(−s).
+(2)
+On the other hand, according to the BGS conjecture, in
+the case of a classically chaotic system, the spectral prop-
+erties must follow the universal statistics of the eigen-
+values of Gaussian random matrices [7]. Several recent
+works have improved the turnover of the BGS conjecture
+in a theorem [11–14]. These proofs still have controver-
+sies and limitations pointed out by some authors [15, 16].
+The terminology ”BGS conjecture” fits the current arti-
+cle for quantized billiards.
+More recently, in [17], the
+conjecture was extended to purely ergodic systems. If
+one disregards spin, in the presence (absence) of time-
+reversal symmetry, p(s) must correspond to that of the
+GOE, Gaussian Orthogonal Ensemble (GUE, Gaussian
+Unitary Ensemble):
+�
+pGOE(s) = (π/2)s exp(−πs2/4),
+pGUE(s) = (32/π2)s2 exp(−4s2/π).
+(3)
+Based on these assumptions, Leyvraz, Schmit, and Selig-
+man (LSS) [18] predicted and tested numerically that
+chaotic billiards with only a three-fold (C3) symmetry
+(without reflection symmetry) have doublets with spec-
+tral statistics of the GUE type, although billiards are
+by time reversal. LSS considered a billiard consisting of
+three straight segments of an equilateral triangle with
+rounded corners by two circumferences of different radii,
+here called Circular-C3 Billiard (C-C3B). In particular,
+LSS showed results for a double ratio between the radii
+where there is a satisfactory agreement for p(s) with the
+GUE statistics, for a total of approximately 800 dou-
+blets. Later, C. Dembowski et al. used microwave bil-
+liards with C3 symmetry to check experimentally the re-
+sult predicted by LSS. Besides, they showed that singlets
+follow GOE [19].
+The BT [20], BGS [21–23] and LSS conjectures have
+been investigated in the literature, but there have been
+comparatively fewer studies on the LSS findings [24–27].
+Until now, little has been said about the situations of
+C3 symmetric billiards with mixed classical phase space,
+where chaotic sea and stable KAM-islands coexist. Here,
+we propose to shed light on the quantum properties of bil-
+liards with mixed classical phase space. For this, we per-
+form numerical calculations on the energy spectra of two
+bi-parametric families of billiards with elliptical sectors
+on their boundaries and analyze the short correlations.
+The first one is the Elliptical Stadium Billiard (ESB), a
+perturbation of the Bunimovich Stadium, whose mixed
+classical phase space was studied in [28, 29]. In sequence,
+we introduce a perturbation of the C-C3B, replacing the
+circumferences with ellipses. Recently, billiards with el-
+liptical borders have been studied in other contexts, i.e.,
+in singular potentials [30], in relativistic limits [31], and
+flows that move around chaotic cores [32].
+We start
+the analysis by presenting the billiards, discussing their
+classical dynamics, showing some mixed phase spaces,
+and calculating the fraction of the chaotic sea on these
+phase spaces. Finally, we follow with the quantized bil-
+liards’ spectral properties, investigating the nns distribu-
+tion p(s) with formulas for intermediate quantum statis-
+tics derived for the doublets recently [27].
+II.
+THE BI-PARAMETRIC BILLIARDS
+FAMILIES AND CLASSICAL DYNAMICS
+The billiards systems studied in this work belong to
+two bi-parametric families, the Elliptical Stadium Bil-
+liards (ESB) and Elliptical-C3 Billiards (E-C3B). The
+first one consists of a perturbation of the Bunimovich
+Stadium. It comprises two half-ellipses (major semi-axis
+a and minor semi-axis 1) that bracket a rectangular sec-
+tor of thickness 2t and height 2.
+[28] showed that in
+the region a ∈ (1,
+√
+2) and t ∈ (0, ∞) are possible to
+find chaotic dynamics or a mixed phase space depending
+on the parameters.
+In [29] is presented a critical be-
+havior of the billiard dynamics near a transition curve,
+t(a) =
+√
+a2 − 1 for the interval a ∈ (1,
+�
+4/3). Based
+on these previous works, we focus our analysis on this
+last interval and t ∈ (0, 1/
+√
+3). The E-C3B is based on
+C-C3B, but ellipses instead of circumferences curve the
+corners. The larger (smaller) ellipse has Ae (ae) and Be
+(be) semi-axes. In all cases described here, the relations
+Ae = 2ae and Be = 2be are maintained, with ae and be in
+the range (0,
+√
+3/6). The LSS billiard is reproduced with
+ae = be =
+√
+3/12. Here, by our knowledge, we present
+for the first time a perturbation on the C-C3B resulting
+in a system that shows a mixed phase space.
+
+3
+A Fundamental Domain (FD) is a neighborhood in Ω
+that contains only one image for any point in the sys-
+tem. Besides the boundary of the ∂Ω, there are addi-
+tional boundaries between adjacent FDs, which are the
+symmetry lines. Classically, billiard dynamics can always
+be reduced to a FD by assuming specular reflections at
+the symmetry lines [33, 34]. For this, we use the FD of
+each billiard in our calculations on the classical dynam-
+ics. In Fig. 1, we graph billiards in families indicating
+the parameters and their respective FDs.
+ESB
+E-C3B
+1
+a
+t
+1
+a
+t
+Symmetry Lines
+Additional Boundaries
+Original Boundaries
+�3/3
+�3/3
+ae
+be
+Ae
+Be
+120º
+120º
+120º
+120º
+(a)
+(b)
+FIG. 1. (a) Original Boundaries of Elliptical Stadium Billiard
+and Elliptical-C3 Billiard. For ESB, the symmetry lines are
+referent to reflections on vertical and horizontal axes. While
+E-C3B are referent to 120◦ rotational axes. (b) Fundamental
+Domains of ESB and E-C3B. The symmetry lines are replaced
+by additional boundaries forming the planar region where we
+analyze these billiards’ classical dynamics.
+The global dynamical properties of the ESB with unit
+mass and velocity may be characterized through colli-
+sions of orbits with the vertical side of its FD shown in
+Fig. 1. An additional part of the boundary dictates this
+edge and does not change with the variation of parame-
+ters. The reduced phase space is then a rectangle defined
+by the vertical position y, where a collision occurs at dis-
+crete time n, and the tangent component of the velocity
+in a collision, vy, with 0 < y < 1 and −1 < vy < 1.
+The small gray dots in Fig.
+2 show the phase plane
+for some values of parameters (a, t) after n = 105 colli-
+sions from the initial conditions (ICs), clearly exhibiting
+a mixed (regular-irregular) characteristic. We plot one
+example of a stable trajectory in red for each one. Quan-
+titative characterization of these mixed-phase spaces can
+be made through the chaotic (regular) fraction ρc (ρr)
+of each phase portrait with ρc + ρr = 1 and 0 ⩽ ρc ⩽ 1.
+The phase plane is partitioned into Nc small disjoint cells
+to measure these quantities [27, 29, 35–37]. For a given
+orbit, let N(n) be the number of different cells in the
+phase space, which are visited up to n impacts in the
+cross-section. The relative measure r(n) is defined as the
+fraction of visited cells averaged over a set of ICs, i.e.,
+r(n) = ⟨N(n)⟩/Nc. So the chaotic fraction of the phase
+space is obtained via
+ρc = lim
+n→∞ lim
+Nc→∞ r(n),
+(4)
+for ICs in the chaotic sea. In our numerical approach,
+we consider Nc = 106, n = 2 · 107, and averages in 20
+random ICs. For the billiards with mixed phase space
+in Fig.
+2, ρ(a)
+c
+= 0.991884 and ρ(b)
+c
+= 0.857184. The
+left panel of Fig.
+3 shows a numerical diagram of ρc.
+The ergodic property (ρc = 1) is numerically guaranteed
+in black regions. This diagram also supports previous
+works [28, 29], where a critical transition from a mixed
+phase space to a fully ergodic was found to cross a critical
+line t(a) =
+√
+a2 − 1.
+y
+y
+vy
+(a) a = 1.04
+      t = 0.15
+(b) a = 1.04
+       t = 0.01
+FIG. 2. Upper Panels: ESB boundaries for some values of pa-
+rameters (a, t) with stable trajectories in red. Lower Panels:
+corresponding phase portraits for 105 collisions with the ver-
+tical boundary from the IC (y0, vy0) = (0.5, 0.0) (small gray
+dots). The red plots correspond to the trajectories in the up-
+per panels. The mixed phase spaces present ρ(a)
+c
+= 0.991884
+and ρ(b)
+c
+= 0.857184.
+The E-C3B’s classical dynamical properties will be
+studied in the same way but are characterized through
+the collisions of the orbits with the horizontal side of its
+FD shown in Fig. 1, which does not change with the vari-
+ation of parameters. The reduced phase space is then a
+rectangle defined by the horizontal position x, and the
+tangent component of the velocity in a collision, vx, with
+0 < x <
+√
+3/3 and −1 < vx < 1. The small gray dots
+in Fig. 4 show the phase plane for some values of pa-
+rameters (ae, be) after n = 2 · 107 collisions from the ICs,
+clearly exhibiting mixed (regular-irregular) characteris-
+tic. The values of chaotic fraction are ρ(a)
+c
+= 0.935152
+and ρ(b)
+c
+= 0.800792. The right panel of Fig. 3 shows a
+numerical diagram of ρc. The ergodic property (ρc = 1)
+is numerically guaranteed in black regions.
+This map
+
+1.0
+0.5
+0.0
+-0.5
+-1.0
+0.2
+0.2
+0.0
+0.4
+0.6
+0.8
+0.4
+0.6
+0.8
+1.0
+0.0
+1.04
+ae
+be
+FIG. 3. Left Panel: diagram of the chaotic fraction of the
+phase space ρc for the ESB. The tiny green line is the critical
+line t(a) =
+√
+a2 − 1 studied in [28, 29]. Right Panel: same
+diagram for the E-C3B showing a distinguished phase space
+behavior depending on the parameters. The ergodic property
+(ρc = 1) is numerically guaranteed in black regions. These
+maps will guide us in exploring quantum properties, where
+these values will be relevant parameters to our analysis.
+will guide us in exploring quantum properties described
+in the next section, where these values will be relevant
+parameters to our analysis.
+G
+x
+x
+(a) ae = 0.2784
+(b) ae = 0.2886088
+vx
+be = 0.256
+be = 0.281522
+FIG. 4.
+Upper Panels: E-C3B boundaries for some values
+of parameters (ae, be) with stable trajectories in red. Lower
+Panels: corresponding phase portraits for 2·107 collisions with
+the horizontal boundary from the IC (x0, vx0) = (0.5, 0.0)
+(small gray dots). The red plots correspond to the trajectories
+in the upper panels. The mixed phase spaces present ρ(a)
+c
+=
+0.935152 and ρ(b)
+c
+= 0.800792.
+III.
+QUANTIZATION AND EIGENVALUES
+SHORT CORRELATIONS
+All Energy spectra {En} of eq.
+(1) were calculated
+with an algorithm based on the scaling method intro-
+duced by E. Vergini and M. Saraceno (VS) in [38]. This
+approach allows us to access high-lying energy eigenval-
+ues that have been unfolded to obtain a unit mean spac-
+ing (⟨sn⟩ = 1) for each billiard. Our results are based
+on sets of approximately 70,000 eigenvalues for a given
+pair of parameters. According to [6], there is possibly no
+more intensely studied spectral statistics more than p(s),
+the density of probability of finding two levels nearest
+neighbor spaced by s.
+A.
+The Singlets Case
+Initially, we focused on results for ESB. Some pro-
+poses have been made to describe these distributions for
+systems whose present mixed-phase space on its classi-
+cal counterpart. Here we focus on two of them. They
+result in intermediate formulas between Poisson and
+GOE statistics through parameters variation.
+Firstly,
+we cite the purely phenomenologic approach by Brody
+[39], where an exponent ν is gradually varied to obtain
+a smooth change between the integrable (ν = 0) and
+chaotic (ν = 1) cases:
+pB(s) = aν(ν + 1)sν exp
+�
+−aνs(ν+1)�
+,
+(5)
+where aν =
+�
+Γ
+�
+ν+2
+ν+1
+��ν+1
+and Γ(x) is the Gamma func-
+tion.
+The second distribution cited here is the Berry-
+Robnik-Brody (BRB), a proposal that takes under con-
+sideration the chaotic (regular) fraction of the classical
+phase space ρc (ρr) [40]:
+pBRB(s) =
+exp(−ρrs)
+
+
+
+ρ2
+r
+(β + 1)Γ
+�
+β+2
+β+1
+�Q
+�
+1
+β + 1; aβ(ρcs)β+1
+�
++
+[2ρrρc + (β + 1)aβρβ+2
+c
+sβ] exp[−aβ(ρcs)β+1]
+�
+.
+(6)
+As in the Brody distribution, aβ =
+�
+Γ
+�
+β+2
+β+1
+��β+1
+and
+Q(κ; x) is the Incomplete Gamma function.
+This dis-
+tribution can go through other distributions varying the
+free parameters ρc and β. For β = 0, pBRB(s) = pP(s)
+and for β = 1 it recovers the distribution of Berry-Robnik
+(BR) [41]. If ρc = 0, pBRB(s) = pP(s) again, while for
+ρc = 1, pBRB = pB(s).
+The nns for ESB were previously studied in [22] with
+around 3,000 eigenvalues of eq.
+(1).
+We use the VS
+method to obtain around 65,000 eigenvalues beyond the
+first 5,000.
+The BRB distribution can fit all p(s) ob-
+tained for all parameters tested on ESB. We have two
+independent parameters for this distribution, ρc, and β.
+However, we fixed ρc at the value obtained in the diagram
+of Fig. 3. The upper panels of Fig. 5 shows representa-
+tive results. The chaotic case presents β = 1.000 ± 0.020,
+the GOE distribution. The mixed (0 < ρc < 1) present
+β = 0.978 ± 0.018 and β = 0.191 ± 0.014, intermediate
+distributions between Poisson and GOE. These results go
+in the direction of the quantum localization, previously
+studied in other billiards systems [42, 43] and discussed
+next.
+
+1.0
+0.5
+0.0
+-0.5
+0.2
+0.0
+0.1
+0.4
+0.5
+0.3
+0.60.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.61.00
+0.3
+0.2
+ae
+0.1
+0.0
+0.78
+0.0
+0.1
+0.2
+0.3
+be1.00
+0.6
+0.5
+0.4
+t 0.3
+0.2
+0.1
+0.0
+1.00
+1.04
+1.08
+1.12
+1.16 0.60
+a5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+1.0
+2.0
+3�
+�
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+1.0
+2.0
+��
+�
+0.0
+1.0
+2.0
+���
+p(s)
+p(s)
+s
+s
+s
+(
+
+ 
+ESB
+E-C3B
+E-C3B
+E-C3B
+ESB
+ESB
+(b)
+ 
+
+ 
+(e)
+ 
+ae = 0.2784 
+be = 0.256
+ae = 0.2
+be = 0.25
+t = 0.287 
+t = 0.15 
+t = 0.01 
+ae = 0.2886088
+be = 0.281522
+FIG. 5. Representative results for BRB distributions fits for p(s). Upper Panels: results on ESB with a = 1.04 and some values
+of t. The chaotic case t = 0.287 (ρc = 1) presents β = 1.000 ± 0.020, the GOE distribution. The mixed cases t = 0.15 and
+t = 0.01 (0 < ρc < 1) present β = 0.978 ± 0.018 and β = 0.191 ± 0.014 respectively, intermediate distributions between Poisson
+and GOE. Lower Panels: results on E-C3B with some values of (ae, be). The chaotic case, (ae, be) = (0.2, 0.25) (ρc = 1) presents
+β = 1.000 ± 0.097, the GOE distribution. The mixed cases, (ae, be) = (0.2784, 0.256) and (ae, be) = (0.2886088, 0.281522)
+(0 < ρc < 1) present β = 0.999 ± 0.057 and β = 0.203 ± 0.073 respectively, in the range of intermediate distributions between
+Poisson and GOE. The fits with the Brody formula and BRB distribution are indistinguishable in both billiards families.
+Quantum dynamical localization corresponds to a pe-
+culiar quantum distribution of the linear or angular mo-
+mentum peaked at zero, with walls that decay exponen-
+tially, differently from the classical results, which pre-
+dicts, for a chaotic or disordered system, a diffusive trans-
+port [44]. The phenomenon can be reviewed in [45]. An
+interesting feature of the quantum dynamical localization
+is that it allows us to estimate the conditions under which
+the comparison with the standard random matrix theory
+is adequate or, in other words, whether an energy eigen-
+values data set belongs to the deep semiclassical regime.
+We follow closely [42] in the short description below. The
+key idea is to express the ergodic parameter α = tH/tT,
+where tH is the (quantum) Heisenberg time, and tT is the
+(classical) transport time, in terms of accessible magni-
+tudes, such as the (quantum) energy E and the (classical)
+number of collisions off the billiard border, NT. From [42]
+the ratio is expressed as
+α = kL
+πNT
+,
+(7)
+where L is the perimeter of the boundary and k2 ∼ E.
+The condition for quantum dynamical localization in a
+given energy spectrum, α ⩽ 1, can then be written as
+k ⩽ kc = πNT/L. To estimate NT, we consider an en-
+semble of orbits initially directed perpendicularly to ∂Ω
+and follow its random spreading as a function of the dis-
+crete time n. The symbols in Fig. 6 illustrate the results
+for the mean square momentum ⟨p2⟩ as a function of
+n in a monolog scale (averaged in sets of 103 randomly
+chosen ICs) for members of two billiards family.
+Sat-
+uration of ⟨p2⟩ occurs at different times NT depending
+on parameters. For the ESB family, all calculated spec-
+tra have kmax ≲ kc as the largest eigenvalue, equivalent
+to the 70,000th level at least. These facts are in agree-
+ment with the intermediate statistics well fitted with eq.
+(6) as in [23, 27, 40, 42]. The same occurs for the sin-
+glets in the E-C3B family, where the condition kmax ≲ kc
+is equivalent to the 70,000th level. The representative
+results are in the lower panels of Fig. 5. The chaotic
+case presents β = 1.000 ± 0.097, the GOE distribution.
+The mixed (0 < ρc < 1) present β = 0.999 ± 0.057 and
+β = 0.203 ± 0.073, in the range of intermediate distribu-
+tions between Poisson and GOE. In the next section, we
+discuss the doublets subspace.
+B.
+The Doublets Case
+Consider a classically chaotic system with time-
+reversal (TR) invariance and a point-group (PG) symme-
+try. If the TR and the PG operations do not commute,
+non-self-conjugate invariant subspaces of the PG must
+exhibit GUE spectral fluctuations instead of GOE ones
+[18]. For example, consider a billiard in the xy plane with
+the C3 symmetry. Such a billiard has eigenfunctions ϕm
+(m = −1, 0, +1), such that ϕ0 is symmetric and repeats
+itself after a rotation of 2π/3 about the symmetry axis,
+whereas ϕ±1 will be repeated only after three consecutive
+rotations of 2π/3. In other words, if R(2π/3) is the rota-
+tion operator for an angle of 2π/3, one has R(2π/3)ϕm =
+exp(i 2π
+3 m)ϕm. Let Θ be the time reversal operator. Θ is
+an antiunitary operator that commutes with the Hamil-
+tonian H, which has eigenvalue Em, i.e., Hϕm = Emϕm.
+It follows that HΘϕm = ΘHϕm = EmΘϕm (Θϕm is also
+an eigenfunction of H with the same eigenvalue Em). Are
+
+6
+0.0
+1.0
+2.0
+ 
+4.0
+5.0
+0.0
+0.1
+0.2
+0
+ 
+0.4
+log10 n
+Elliptical-C3 Billiard
+Elliptical Stadium Billiard
+0.0
+0.1
+0.2
+0.3
+0.4
+FIG. 6. Calculated mean square of the momentum as a func-
+tion of the discrete time n in a monolog scale (number of
+collisions of the particle off the billiard boundary). Lines are
+guides for the eyes. Upper panel: results for members of the
+ESB family with a = 1.04. The red dots are for t = 0.287
+and present saturation at NT ≃ 7.102.
+Blue dots are for
+t = 0.15, and saturation at NT ≃ 2.103, and black dots are for
+t = 0.01 presenting NT ≃ 6.102. Lower Panel: same calcula-
+tions for the E-C3B, the red dots are for (ae, be) = (0.2, 0.25)
+and present saturation at NT ≃ 3.102.
+Blue dots are for
+(ae, be) = (0.2784, 0.256) and saturation at NT ≃ 7.102, and,
+black dots are for (ae, be) = (0.2886088, 0.281522) presenting
+NT ≃ 2.103.
+ϕm and Θϕm the same eigenstate? For this subspace one
+may write Θϕm = (−1)mϕ−m. Thus, Θϕ0 = ϕ0, i.e., ϕ0
+is a singlet. The top panels in Fig. 7 show cases of the
+probability density |varphi0|2. On the other hand, ϕ1
+and ϕ−1 must correspond to distinct states. One refers
+to this doublet state as a Kramers degeneracy. The mid-
+dle panels in Fig. 7 show the real and imaginary parts
+of the member ϕ1 of a doublet, say (ϕ1, ϕ−1), in the
+same billiard. The probability density |ϕ1|2 recovers the
+C3 symmetry (rightmost middle panel in Fig. 7). The
+bottom panels in Fig. 7 show the same state under the
+application under rotation operator R(2π/3). A complex
+conjugation of the shown state obtains the other mem-
+ber ϕ−1 of the doublet.
+Since these degenerate states
+are not TR invariant, they must follow the GUE of ran-
+dom matrices, providing the billiard is classically chaotic,
+according to the LSS results.
+For the E-C3B, the degenerate states remain invariant
+to TR. However, the spectral distribution will be changed
+for cases where the classical dynamics is not completely
+chaotic (ρc < 1), with a p(s) resultant that deviates from
+the GUE case. Thus, it is necessary to use new inter-
+mediate formulas to study the distribution of doublets in
+billiards with mixed classical phase space. The following
+formulas we derived in [27].
+Following the same steps
+in [39] led to the eq. (5), a Brody-like formula for the
+transition between the Poisson and GUE distributions is
+FIG. 7. Top panels: Density plots of squared eigenfunctions
+corresponding to singlet states in the LSS billiard, exhibiting
+the underlying C3 symmetry. In the color scale, |ϕ1,a|2 is the
+maximum probability in each case. Middle panels: Real and
+imaginary parts of a member ϕ1 of a doublet. In the color
+scale, ±ϕ1,a is the minimum and maximum of the wave func-
+tion. The probability density recovers the C3 symmetry (right
+panels). Bottom panels: Same state in the middle under the
+application of the rotation operator R(2π/3).
+obtained, namely,
+pB,2(s) = (η + 1)b2
+ηs2η exp
+�
+−bηsη+1�
+,
+(8)
+where
+bη =
+�
+Γ
+�2η + 1
+η + 1
+��−(η+1)
+,
+(9)
+and 0 ⩽ η ⩽ 1. For η = 0, pB,2(s) reduces to the Poisson
+distribution, whereas for η = 1, the Wigner distribution
+for the GUE is obtained. In [40], the dynamical local-
+ization of chaotic eigenstates was taken into account and
+their coupling with the regular ones through tunneling
+effects. The so-called BRB distribution previously dis-
+cussed in sec.
+III A. Following this, the formula that
+corresponds to the Poisson ↔ GUE crossover is
+pBRB,2(s)eρrs =
+ρrρcb
+1
+γ+1
+γ
+(2 − ρrs) Q
+�1 + 2γ
+1 + γ ; bγ (ρcs)γ+1
+�
++
+�
+ρ2
+r
+�
+1 + bγργ+1
+c
+sγ+1�
++
+(1 + γ)
+�
+ργ+1
+2
+bγsγ�2 �
+e−bγ(ρcs)γ+1,
+(10)
+where bγ is defined as in eq. (9) and Q(κ; x) is the incom-
+plete Gamma function. Here, pBRB,2(s) = pP(s) if ρr = 1
+or if γ = 0, and pBRB,2(s) = pB,2(s) if ρr = 0. In [27], the
+above formula was widely tested only in the regime of
+
+1.5
+1
+0.5
+0
+-0.5
+-1
+-1.5
+-24
+3.5
+3
+2.5
+2
+1.5
+1
+0.507
+full ergodicity (polygonal cases) and in a single case with
+ρc < 1. Here, we detail a non-polygonal billiards family
+that produces a wide variability of ρc values. In these
+cases, pBRB,2 well-fitted distributions of nns for ρc < 1
+for all investigated cases. The representative results are
+in Fig. 8. As in the previous section, the doublets sub-
+space is in the region of the spectrum such that k ≲ kc,
+equivalent to 60,000th level.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+E-C3B
+E-C3B
+E-C3B
+(a)
+(b)
+
+ae = 0.2784
+be = 0.256
+ae = 0.2
+be = 0.25
+ae = 0.2886088
+be = 0.281522
+p(s)
+p(s)
+p(s)
+s
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+1.0
+2.0
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FIG. 8. Representative results for BRB-like distributions, eq.
+(10), fits for p(s) in doublets subspace for same members of E-
+C3B family of Fig. 5. In panel (a), the chaotic case (ae, be) =
+(0.2, 0.25) (ρc = 1) presents γ = 0.960 ± 0.050, in the range
+of a GUE distribution. In panels (b) and (c), the mixed cases
+(ae, be) = (0.2784, 0.256) and (ae, be) = (0.2886088, 0.281522)
+(0 < ρc < 1) present γ = 1.000 ± 0.032 and γ = 1.00 ±
+0.13 respectively, in the range of intermediate distributions
+between Poisson and GUE. Fits with Brody-like formula (8),
+and BRB-like distribution (10), are indistinguishable.
+IV.
+CONCLUSIONS ANS PERSPECTIVES
+This paper presents numerical results on classical dy-
+namics and quantization in two bi-parametric billiard
+families. The ESB comprises two ellipses of minor semi-
+axe unitary, major semi-axe a, and a rectangular region of
+length 2t [28, 29]. The other family, introduced here as E-
+C3B, presents the C3 symmetry [18, 25, 27] and is formed
+by an equilateral triangle with rounded corners by two
+ellipses with semi-axis Ae = 2ae and Be = 2be. First, we
+investigate the classical dynamics of these billiards where
+we built detailed diagrams for the chaotic fraction (ρc) of
+their phase spaces. After that, we investigated the nns
+distributions p(s) for these systems, a measure of short
+correlations. In the asymmetric ESB family, the param-
+eters space region (a, t) where the classical phase space
+is mixed (regular and chaotic regions coexist), all found
+statistics present intermediated results between Poisson
+and GOE distributions. The BRB distribution [40], eq.
+(6), very well fitted all cases.
+These results perfectly
+agree with the expected from the ergodic parameter α
+that signals the possibility of quantum dynamical local-
+ization when α < 1. All sets of eigenvalues used as data
+are in a range of energy that satisfies this condition. In
+the E-C3B family, the eigenstates can be split into sin-
+glets and doublets subspaces due the symmetry. The first
+subspace presents similar results to the previous family,
+reinforcing the agreement with the expected energy range
+set with α < 1 [43]. The doublets subspace, whose for
+the chaotic cases is expected a GUE distribution shows
+the more relevant result in this work. All found statistics
+present intermediated results between Poisson and GUE
+distributions for the parameter space (ae, be) where the
+classical phase space is mixed. A BRB-like formula [27],
+eq. (10), well fitted all cases. This formula was tested for
+ρc < 1 and α < 1 in just a few cases. Particularly in the
+E-C3B family, the minimum value of the chaotic fraction
+of the classical phase space is ρc ≃ 0.8. This limitation
+can be avoided if we set free the conditions Ae = 2ae
+and Be = 2be, used here to follow closer to the C-C3B
+introduced by LSS. In this perspective, a phase diagram
+analog to Fig. 3 even more intricate is generated, possible
+further explorations of eq. (10).
+The parameter β in eq. (6) was extensively compared
+with other localization metrics, including analyses involv-
+ing Husimi functions, calculations of the entropy localiza-
+tion measure [42], and normalized inverse participation
+ratio [23]. How the new distribution, eq. (10), uses the
+same arguments to include the parameter γ is merito-
+rious in a future comparison between this quantity and
+other localization metrics.
+Another theme meritorious
+of investigation is the level statistics in an energy range
+that α ≫ 1. The BR formulas are expected to provide
+a good description of the deep semiclassical regime [41],
+an excellent agreement has been found with numerical
+experiments in a billiard for which the eigenvalues set
+is around 1,500,000th level [42], an impressive number.
+The BR-like formula in [27] should be tested in a range
+of high energy in the doublets subspace to close the com-
+parisons between the short correlations in the singlets
+sets and doublets subspace. In addition, our results indi-
+cate an intriguing correlation between singlets and dou-
+blets spectra for the E-C3B family, producing p(s)’s that
+move away from the GOE and GUE distributions as ρc
+decreases, thus requiring a further investigation of the ob-
+
+8
+served effect. In this perspective, a range opens up to in-
+vestigate the correlation of spectra of different subspaces
+[26, 34, 46–48] in billiards that present only rotational
+symmetries greater than three, which will give the pos-
+sibility of performing other tests with the new formulas
+(8) and (10).
+ACKNOWLEDGMENTS
+Useful discussions with F. M. de Aguiar and K. Terto
+are gratefully acknowledged. This work has been sup-
+ported by the Brazilian Agencies CNPq, CAPES and
+FACEPE.
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+Computational analysis of NM-polynomial based topological
+indices and graph-entropies of carbon nanotube Y-junctions
+Sohan Lal1, Vijay Kumar Bhat1,∗, Sahil Sharma1
+1School of Mathematics, Shri Mata Vaishno Devi University,
+Katra-182320, Jammu and Kashmir, India.
+[email protected], [email protected], [email protected]
+Abstract
+Carbon nanotube Y-junctions are of great interest to the next generation of innovative
+multi-terminal nanodevices. Topological indices are graph-theoretically based parameters that
+describe various structural properties of a chemical molecule.
+The entropy of a graph is a
+topological descriptor that serves to characterize the complexity of the underlying molecular
+graph.
+The concept of entropy is a physical property of a thermodynamic system.
+Graph
+entropies are the essential thermophysical quantities defined for various graph invariants and
+are applied to measure the heterogeneity and relative stabilities of molecules. In this paper,
+several neighborhood degree sum-based topological indices including graph-based entropies of
+carbon nanotube Y-junction graphs are computed.
+Keywords: Armchair carbon nanotube, graph entropy, NM-polynomial, topological indices, Y-
+junction graph.
+MSC (2020): 05C10, 05C35, 05C90
+1
+Introduction
+Nanotechnology is currently popular because of its evolving, electron transfer property and low-cost
+implementation.
+Nanotubes [1], were discovered in 1985 and carbon nanotubes [2] in 1991.
+In
+nanoscience and technology, branched or non-straight carbon nanotubes such as L, T, X, and Y
+have a lot of applications in electronic devices, such as three-terminal transistors, multi-terminal
+nanoelectronics, switches, amplifiers, etc., [3, 4, 5, 6, 7, 8]. These junctions are a great option for the
+production of nanoscale electronic devices with better switching and reliable transport properties at
+room temperature. For more applications of carbon nanotube Y-junctions, we refer to [9, 10, 11].
+The first proposed branched carbon nanotube was of Y shape, commonly known as Y-junction or
+three-terminal junction. These junctions are classified as an armchair, zig-zag, or chiral depending
+on the chirality of connected carbon nanotubes. Also, they can be single-walled or multi-walled,
+symmetric or asymmetric, capped or uncapped. A carbon nanotube is called uncapped if both ends
+are open. A Y-junction is called symmetric if the nanotubes joining in the Y shape are identical,
+heptagons appeared isolated, and are distributed symmetrically. For various symmetric and asym-
+metric carbon nanotube Y-junctions, we refer to [12, 13, 14, 15].
+A carbon nanotube Y-junction is formed by joining three identical carbon nanotubes in a Y-
+shaped pattern. These junctions contain exactly six hexagons as well as heptagons at the branch-
+ing points. The first structural model of symmetrical single-walled armchair carbon nanotube Y-
+junctions was proposed by Chernozatonskii [16] and Scuseria [17], independently, in 1992. These
+junctions were experimentally observed [18] in 1995. For more applications and properties of carbon
+nanotube Y-junction graphs, we refer to [19, 20, 21].
+Mathematical chemistry is a branch of theoretical chemistry that employs mathematical tech-
+niques to explain the molecular structure of a chemical molecule and its physicochemical properties.
+Molecular graphs are a visual representation of a chemical molecule with vertices representing atoms
+and edges representing bonds between the atoms [22]. Let G = (V (G), E(G)) be a molecular graph
+with vertex set V (G) and edge set E(G). The order of a molecular graph G is defined as the total
+number of vertices in G, denoted by |V (G)|, and the number of edges in G is called size of G, denoted
+by |E(G)|. Any edge of the graph connecting its vertices u and v, is denoted by e = uv ∈ E(G).
+Two vertices of graph G are said to be adjacent if there exists an edge between them. The degree
+of vertex v ∈ V (G), denoted by d(v), is defined as the number of vertices that are adjacent to
+1
+arXiv:2301.02169v1  [cond-mat.mes-hall]  3 Jan 2023
+
+vertex v, i.e., d(v)= |{u : e = uv ∈ E(G)}|. The neighborhood degree sum of vertex v ∈ V (G) is
+denoted by dn(v), and is defined as the sum of the degrees of all vertices that are adjacent to v,
+i.e., dn(v) = �
+u
+d(v): uv ∈ E(G). The minimum cardinality of the set K ⊆ V (G) such that G \ K
+is disconnected graph is called connectivity or vertex-connectivity of a connected graph G. The
+connected graph G is said to be k-connected if its connectivity is k.
+Topological indices are the numerical values calculated from molecular graphs to describe various
+structural properties of the chemical molecule. They are frequently used to model many physico-
+chemical properties in various quantitative structure-property/activity relationship (QSPR/QSAR)
+studies [23, 24, 25]. In 1947, the chemist Harold Wiener [26] initiated the concept of topological
+indices. Since then, various topological indices have been introduced, and a lot of research has been
+conducted toward computing the indices for different molecular graphs and networks. A topological
+index based on the degree of end vertices of an edge can predict various physicochemical properties
+of the molecule, such as heat of formation, strain energy, entropy, enthalpy, boiling points, flash
+point, etc., without using any weight lab [24].
+The Zagreb indices and their variations have been used to investigate molecular complexity, ZE-
+isomerism, and chirality [27]. In general, the Zagreb indices have shown applicability for deriving
+multilinear regression models. Ghorbani and Hosseinzadeh [28] introduced the third version of the
+Zagreb index and shows that this index shows a good correlation with acentric factor and entropy
+of the octane isomers. Mondal et al. [29] introduced neighborhood degree sum-based topological
+indices namely neighborhood version of forgotten topological index and neighborhood version of
+second modified Zagreb index and discuss some mathematical properties and degeneracy of these
+novel indices. For more neighborhood degree sum-based topological indices, their properties, and
+applications, we refer to [24, 30, 31].
+The process of computing the topological indices of a molecular graph from their definitions is
+complex and time-consuming. Thus, for a particular family of graphs and networks, algebraic poly-
+nomials play an important role in reducing the computational time and complexity when computing
+its topological indices. In short, with the help of algebraic polynomials, one can easily compute
+various kinds of graph indices within a short span of time. The NM-polynomial plays vital role in
+the computation of neighborhood degree sum-based topological indices. Let dn(v) denotes the neigh-
+borhood degrees sum of vertex v ∈ V (G). Then, the neighborhood M-polynomial (NM-polynomial)
+of G is defined as [30, 32, 33]
+NM(G; x, y) =
+�
+i≤j
+|Eij(G)|xiyj
+(1)
+where, |Eij(G)|, i, j ≥ 1, be the number of all edges e = uv ∈ E(G) such that {dn(u) = i, dn(v) = j}.
+Recently, various neighborhood degree sum-based topological indices have been computed via
+the NM-polynomial technique. For example, Mondal et al. [30, 34] obtained some neighborhood
+and multiplicative neighborhood degree sum-based indices of molecular graphs by using their NM-
+polynomials. Kirmani et al. [24] and Mondal et al. [35], investigated some neighborhood degree
+sum-based topological indices of antiviral drugs used for the treatment of COVID-19 via the NM-
+polynomial technique. Shanmukha et al. [36] computed the topological indices of porous graphene
+via NM-polynomial method. For more neighborhood degree sum-based topological indices via NM-
+polynomials, we refer to [24, 35, 37, 38].
+Some neighborhood degree sum-based topological indices and their derivation from NM-polynomial
+are given in Table 1.
+In chemical graph theory, the determination of the structural information content [39] of a graph
+is mostly based on the vertex partition of a graph to obtain a probability distribution of its vertex
+set [40]. Based on such a probability distribution, the entropy of a graph can be defined. Thus,
+the structural information content of a graph is defined as the entropy of the underlying graph
+topology. The concept of graph entropy or entropy of graph was first time appeared in [41], where
+molecular graphs are used to study the information content of an organism. Entropy-based methods
+are powerful tools to investigate various problems in cybernetics, mathematical chemistry, pattern
+recognition, and computational physics [22, 39, 42, 43, 44].
+2
+
+Table 1: Description of some topological indices and its derivation from NM-polynomial
+Topological index
+Formula
+Derivation from NM(G; x, y)
+Third version of Zagreb index [28]: NM1(G)
+�
+uv∈E(G)
+�
+dn(u) + dn(v)
+�
+(Dx + Dy)(NM(G; x, y))|x=y=1
+Neighborhood second Zagreb index [29]: NM2(G)
+�
+uv∈E(G)
+�
+dn(u)dn(v)
+�
+(DxDy)(NM(G; x, y))|x=y=1
+Neighborhood second modified Zagreb index [30]: nmM2(G)
+�
+uv∈E(G)
+�
+1
+dn(u)dn(v)
+�
+(SxSy)(NM(G; x, y))|x=y=1
+Neighborhood forgotten topological index [29]: NF (G)
+�
+uv∈E(G)
+�
+d2
+n(u) + d2
+n(v)
+�
+(D2
+x + D2
+y)(NM(G; x, y))|x=y=1
+Third NDe index [30]: ND3(G)
+�
+uv∈E(G)
+dn(u)dn(v)(dn(u) + dn(v))
+DxDy(Dx + Dy)(NM(G; x, y))|x=y=1
+Neighborhood general Randic index [30]: NRα(G)
+�
+uv∈E(G)
+dα
+n(u)dα
+n(v)
+(Dα
+x Dα
+y )(NM(G; x, y))|x=y=1
+Neighborhood inverse Randic index [30]: NRRα(G)
+�
+uv∈E(G)
+1
+dα
+n(u)dα
+n(v)
+(Sα
+x Sα
+y )(NM(G; x, y))|x=y=1
+Fifth NDe index [30]: ND5(G)
+�
+uv∈E(G)
+� d2
+n(u)+d2
+n(v)
+dn(u)dn(v)
+�
+(DxSy + SxDy)(NM(G; x, y))|x=y=1
+Neighborhood harmonic index [30]: NH(G)
+�
+ab∈E(G)
+2
+dn(u)+dn(v)
+2SxT (NM(G; x, y))|x=y=1
+Neighborhood inverse sum indeg index [30]: NI(G)
+�
+uv∈E(G)
+� dn(u)dn(v)
+dn(u)+dn(v)
+�
+(SxT DxDy)(NM(G; x, y))|x=y=1
+where, Dx = x
+� (∂(NM(G;x,y))
+∂x
+�
+, Dy = y
+� (∂(NM(G;x,y))
+∂y
+�
+, Sx =
+� x
+0
+NM(G;t,y)
+t
+dt, Sy =
+� y
+0
+NM(G;x,t)
+t
+dt,
+T (NM(G; x, y)) = NM(G; x, x).
+Entropy is a measure of randomness, uncertainty, heterogeneity, or lack of information in a sys-
+tem. Based on information indices, there are various approaches to deriving graph entropy from the
+topological structure of a given chemical molecule [45]. For example, Trucco [39] and Rashevsky
+[41] defined graph entropies in terms of degree of vertex, extended degree sequences, and number of
+vertices of a molecular graph. Tan and Wu [46] study network heterogeneity by using vertex-degree
+based entropies. Mowshowitz defined the entropy of a graph in terms of equivalence relations de-
+fined on the vertex set of a graph and discussed some properties related to structural information
+[47, 48, 49, 50].
+Recently, Shabbir and Nadeem [51] defined graph entropies in terms of topological indices for the
+molecular graphs of carbon nanotube Y-junctions and developed the regression models between the
+graph entropies and topological indices. Nadeem et al. [52] calculated some degree-based topological
+indices for armchair carbon semicapped and capped nanotubes and investigated their chemical and
+physical properties. Baˇca et al. [53] computed some degree-based topological indices of a carbon
+nanotube network and studied its properties. Azeem et al. [54] calculated some M-polynomials
+based topological indices of carbon nanotube Y-junctions and their variants. Ahmad [55], studied
+some ve-degree based topological indices of carbon nanotube Y-junctions and discussed their proper-
+ties. Ayesha [56] calculated the bond energy of symmetrical single-walled armchair carbon nanotube
+Y-junctions and developed regression models between bond energy and topological indices. Rahul et
+al. [57] calculated some degree-based topological indices and graph-entropies of graphene, graphyne,
+and graphdiyne by using Shannon’s approach.
+The above-mentioned literature and applications of carbon nanotubes in the field of nanoscience
+and technology inspired us to develop more research on the molecular structure of carbon nanotube
+Y-junction and their variants. In addition, no work has been reported on NM-polynomial based
+topological indices and index-entropies of Y-junction graphs. Therefore, the main contribution of
+this study includes the following:
+• Computation of NM-polynomials of carbon nanotube Y-junction graphs.
+• Computation of some neighborhood degree sum-based topological indices from NM-polynomials.
+• Some graph index-entropies in terms of topological indices are defined and computed.
+3
+
+• Comparative analysis of obtained topological indices and graph index-entropies of Y-junction
+graphs.
+2
+Aim and Methodology
+We use the edge partition technique, graph-theoretical tools, combinatorial computation, and the
+degree counting method to derive our results. The degree of end vertices is used to generate the
+patterns of edge partitions of the Y-junction graphs. Using such partitions, a general expression
+of NM-polynomials is derived. Then, several neighborhood degree sum-based topological indices
+are obtained from the expression of these NM-polynomials with the help of Table 1. Also, graph
+index- entropies in terms of topological indices have been defined by using edge-weight functions
+and computed for Y-junction graphs.
+The paper is structured as follows: In Section 3, we define topological index-based graph en-
+tropies. The Y-junction graphs and their constructions are described in Section 4. In Section 5, the
+general expression of the NM-polynomials and neighborhood degree sum-based topological indices
+of Y-junction graphs are presented. Section 6 describes the graph index-entropies of Y-junction
+graphs. The numerical analysis of the findings is discussed in Section 7. Finally, the conclusion is
+drawn and discussed in Section 8.
+3
+Definitions and Preliminaries
+In this section, we define graph index-entropies in terms of an edge-weight function. In 2008, Dehmer
+[40] defined the entropy for a connected graph G as follows:
+Definition 1. [40] Let G = (V (G), E(G)) be a connected graph of order n and g be an arbitrary
+information functional. Then the entropy of G is defined as
+Hg(G) = −
+n
+�
+i=1
+g(vi)
+n�
+i=1
+g(vi)
+log
+�
+g(vi)
+n�
+i=1
+g(vi)
+�
+.
+(2)
+Since an information function defined on the vertex set of a graph is an arbitrary function. Hence,
+Dehmer’s definition shows the possibility of producing various graph entropies for a variation in the
+selection of information functionals. For such graph entropy, we can refer to [58, 59, 60].
+Let β : E(G) → R+ ∪ {0} be an edge-weight function and dn(u) =
+�
+uv∈E(G)
+d(u), denotes the
+sum of degrees of end vertices of an edges incident to vertex u ∈ V (G) (also known as neighborhood
+degree-sum of vertex u). Then, for eight different edge-weight functions, the third-version of Zagreb
+index, neighborhood second Zagreb index, neighborhood forgotten topological index, neighborhood
+second modified Zagreb index, third NDe index, fifth NDe index, neighborhood harmonic index and
+neighborhood inverse sum indeg index-entropies have been defined in the following manner:
+• Third-version of Zagreb index-entropy: If e = uv is an edge of a connected graph G and
+β1(e) = dn(u) + dn(v) is an edge-weight function defined on E(G). Then, the third-version of
+Zagreb index is
+NM1(G) =
+�
+e=uv∈E(G)
+β1(e) =
+�
+e=uv∈E(G)
+dn(u) + dn(v).
+(3)
+Equation (2) for this edge-weight function gives us
+Hβ1(G)
+=
+−
+�
+e∈E(G)
+β1(e)
+�
+e∈E(G)
+β1(e)log
+�
+β1(e)
+�
+e∈E(G)
+β1(e)
+�
+=
+−
+1
+�
+e∈E(G)
+β1(e)
+�
+e∈E(G)
+β1(e)
+�
+log(β1(e)) − log
+�
+e∈E(G)
+β1(e)
+�
+4
+
+=
+−
+1
+�
+e∈E(G)
+β1(e)
+�
+e∈E(G)
+β1(e)log(β1(e)) +
+1
+�
+e∈E(G)
+β1(e)
+�
+e∈E(G)
+β1(e)log
+�
+�
+e∈E(G)
+β1(e)
+�
+=
+log
+�
+�
+e∈E(G)
+β1(e)
+�
+−
+1
+�
+e∈E(G)
+β1(e)
+�
+e∈E(G)
+β1(e)log(β1(e)).
+On replacing
+�
+e∈E(G)
+β1(e) by NM1(G) in the above equation, we get the following third-version
+of Zagreb index-entropy
+Hβ1(G) = log(NM1(G)) −
+1
+NM1(G)
+�
+e∈E(G)
+β1(e)logβ1(e).
+(4)
+Similarly, we define other graph index-entropies as follows:
+• Neighborhood second Zagreb index-entropy: For β2(e) = dn(u)dn(v), the neighborhood
+second Zagreb index and neighborhood second Zagreb index-entropy are
+NM2(G) =
+�
+e=uv∈E(G)
+dn(u)dn(v),
+(5)
+and
+Hβ2(G) = log(NM2(G)) −
+1
+NM2(G)
+�
+e∈E(G)
+β2(e)logβ2(e).
+(6)
+• Neighborhood forgotten topological index-entropy: For β3(e) = d2
+n(u) + d2
+n(v), the
+neighborhood forgotten topological index and neighborhood forgotten topological index-entropy
+are
+NF(G) =
+�
+e=uv∈E(G)
+d2
+n(u) + d2
+n(v),
+(7)
+and
+Hβ3(G) = log(NF(G)) −
+1
+NF(G)
+�
+e∈E(G)
+β3(e)logβ3(e).
+(8)
+• Neighborhood second modified Zagreb index-entropy: For β4(e) =
+1
+dn(u)dn(v), the
+neighborhood second modified Zagreb index and neighborhood second modified Zagreb index-
+entropy are
+nmM2(G) =
+�
+e=uv∈E(G)
+1
+dn(u)dn(v),
+(9)
+and
+Hβ4(G) = log(nmM2(G)) −
+1
+nmM2(G)
+�
+e∈E(G)
+β4(e)logβ4(e).
+(10)
+• Third NDe index-entropy: For β5(e) = dn(u)dn(v)
+�
+dn(u) + dn(v)
+�
+, the third NDe index
+and third NDe index-entropy are
+ND3(G) =
+�
+e=uv∈E(G)
+dn(u)dn(v)
+�
+dn(u) + dn(v)
+�
+,
+(11)
+and
+Hβ5(G) = log(ND3(G)) −
+1
+ND3(G)
+�
+e∈E(G)
+β5(e)logβ5(e).
+(12)
+• Fifth NDe index-entropy: For β6(e) = dn(u)
+dn(v) + dn(v)
+dn(u), the fifth NDe index and fifth NDe
+index-entropy are
+ND5(G) =
+�
+e=uv∈E(G)
+dn(u)
+dn(v) + dn(v)
+dn(u),
+(13)
+and
+Hβ6(G) = log(ND5(G)) −
+1
+ND5(G)
+�
+e∈E(G)
+β6(e)logβ6(e).
+(14)
+5
+
+• Neighborhood harmonic index-entropy: For β7(e) =
+2
+dn(u)+dn(v), the neighborhood har-
+monic index and neighborhood harmonic index-entropy are
+NH(G) =
+�
+e=uv∈E(G)
+2
+dn(u) + dn(v),
+(15)
+and
+Hβ7(G) = log(NH(G)) −
+1
+NH(G)
+�
+e∈E(G)
+β7(e)logβ7(e).
+(16)
+• Neighborhood inverse sum indeg index-entropy: For β8(e) =
+dn(u)dn(v)
+dn(u)+dn(v), the neighbor-
+hood inverse sum index and neighborhood inverse sum index-entropy are
+NI(G) =
+�
+e=uv∈E(G)
+dn(u)dn(v)
+dn(u) + dn(v),
+(17)
+and
+Hβ8(G) = log(NI(G)) −
+1
+NI(G)
+�
+e∈E(G)
+β8(e)logβ8(e).
+(18)
+4
+Y-Junction Graphs
+The Y-junctions examined in this study are created by the covalent connection of three identical
+single-walled carbon nanotubes crossing at an angle of 120◦ and are uniquely determined by their
+chiral vector v = nv1 +nv2, where v1 and v2 are graphene sheet lattice vectors and n is non-negative
+integer. Let m ≥ 1 and n ≥ 4 be an even integer. Then, an uncapped symmetrical single-walled
+carbon nanotube Y-junction is made up of an armchair Y (n, n) and three identical single-walled
+armchair carbon nanotubes Tm(n, n) each of length m (layers of hexogones), denoted by Y m(n, n).
+In Y m(n, n), we have 3
+4n2 − 3
+2n + 5 faces including three openings (where the tubes meet to the
+amchair) each of chirality (n, n), six heptagones, and 3
+4n2 − 3
+2n − 4 hexagones. In addition, the tube
+Tm(n, n) contains 2mn hexagonal faces.
+Let n, m, and l be positive integers with m ≥ 1 and n = 2l, for some l ≥ 2. Then J = Jm(n, n)
+be the Y -junction graph of Y m(n, n). It has 9l2 − 3l + 2 hexagonal rings along with six heptagons.
+The graph J is of order 6l2 +18l +6+24ml and size 9l2 +21l +9+36ml. It has 6l2 +12l +6+24ml
+vertices of degree three and 12l vertices of degree two. Note that graph J is a 2-conneced graph.
+Along with 2-connected Y-junction graph J, the 1-connected Y-junction graphs have also been
+taken into consideration. These graphs are obtained by adding pendants to the degree 2 vertices
+of the 2-connected graph J. Note that, each tube of J has 2n vertices of degree 2. Therefore, the
+graph J has 6n vertices of degree 2.
+The graph obtained by connecting 2n pendants to any one tube in J is denoted by J1, and we
+call it as second type Y-junction graph. The order and size of graph J1 are 6l2 + 22l + 6 + 24ml and
+9l2 + 25l + 9 + 36ml, respectively. The graph J2 represents a graph which is obtained by attaching
+4n pendants to any two tubes of J and we call it as third type Y-junction graph. In J2, we have
+6l2 + 26l + 6 + 24ml vertices and 9l2 + 29l + 9 + 36ml edges. The graph obtained by joining 6n
+pendants to all the three tubes of J is denoted by J3, and we called it as fourth type Y-junction
+graph. It has 6l2 + 30l + 6 + 24ml vertices and 9l2 + 33l + 9 + 36ml edges. The carbon nanotube
+Y-junction graphs J, J1, J2, and J3 are shown in Figure 1.
+The edge partition of Y-junction graphs J, J1, J2, and J3 based on the neighborhood degree-sum
+of end vertices of an edge is given in Table 2.
+6
+
+(a) Y-junction graph J
+(b) Y-junction graph J1
+(c) Y-junction graph J2
+(d) Y-junction graph J3
+Figure 1: A symmetrical uncapped single-walled armchair carbon nanotubes Y-junction graphs
+Table 2: Edge partitions of J, J1, J2, and J3
+dn(u), dn(v)
+J-frequency
+J1-frequency
+J2-frequency
+J3-frequency
+(3,7)
+0
+4l
+8l
+12l
+(5,5)
+6l
+4l
+2l
+0
+(5,8)
+12l
+8l
+4l
+0
+(7,7)
+0
+2l
+4l
+6l
+(7,9)
+0
+4l
+8l
+12l
+(8,8)
+6l
+4l
+2l
+0
+(8,9)
+12l
+8l
+4l
+0
+(9,9)
+9l2 − 15l + 36ml + 9
+9l2 − 9l + 36ml + 9
+9l2 − 3l + 36ml + 9
+9l2 + 3l + 36ml + 9
+5
+NM-Polynomials and Topological Indices of Y-Junction
+Graphs
+In this section, we develop the general expression of NM-polynomials for the Y-junction graphs and
+then recover various neighborhood degree-sum based topological indices from these polynomials.
+Theorem 1. Let J be the Y-junction graph of an uncapped symmetrical single-walled armchair
+carbon nanotube. Then
+NM(J; x, y) = 6lx5y5 + 12lx5y8 + 6lx8y8 + 12lx8y9 + (9l2 − 15l + 9 + 36ml)x9y9.
+Proof. The Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotubes
+has 9l2 +21l +9+36ml number of edges. Let E(i,j) be the set of all edges with neighborhood degree
+sum of end vertices i, j, i.e., E(i,j) = {uv ∈ E(J) : dn(u) = i, dn(v) = j}.
+7
+
+Extension of J to JiBy means of structural analysis of J, the edge set of J can be partitioned into five sets on the basis
+of neighborhood degree sum of end vertices as follows:
+E(5,5) = {uv ∈ E(J) : dn(u) = 5, dn(v) = 5}, E(5,8) = {uv ∈ E(J) : dn(u) = 5, dn(v) = 8},
+E(8,8) = {uv ∈ E(Jm(n, n)) : dn(u) = 8, dn(v) = 8}, E(8,9) = {uv ∈ E(J) : dn(u) = 8, dn(v) = 9},
+E(9,9) = {uv ∈ E(J) : dn(u) = 9, dn(v) = 9}, and |E(5,5)| = 6l, |E(5,8)| = 12l, |E(8,8)| = 6l,
+|E(8,9)| = 12l, |E(9,9)| = 9l2 − 15l + 9 + 36ml.
+From Equation (1), the NM-polynomial of J is obtained as follows:
+NM(J; x, y)
+=
+�
+i≤j
+|E(i,j)|xiyj
+=
+|E(5,5)|x5y5 + |E(5,8)|x5y8 + |E(8,8)|x8y8 + |E(8,9)|x8y9 + |E(9,9)|x9y9
+=
+6lx5y5 + 12lx5y8 + 6lx8y8 + 12lx8y9 + (9l2 − 15l + 9 + 36ml)x9y9.
+Theorem 2. Let J be the Y-junction graph of an uncapped symmetrical single-walled armchair
+carbon nanotube . Then
+(i) NM1(J) = 162l2 + 246l + 648ml + 162
+(ii) NM2(J) = 729l2 + 663l + 2916ml + 729
+(iii) NF(J) = 1458l2 + 1446l + 5832ml + 1458
+(iv) nmM2(J) = 0.11l2 + 0.62l + 0.44ml + 0.11
+(v) NRα(J) = 6l(25α + 2(40)α + 64α + 2(72)α) + 81α(9l2 − 15l + 9 + 36ml)
+(vi) ND3(J) = 13122l2 + 6702l + 52488ml + 13122
+(vii) ND5(J) = 18l2 + 44.86l + 72ml + 18
+(viii) NH(J) = l2 + 9.69l + 4ml + 1
+(ix) NI(J) = 40.5l2 + 59.24l + 162ml + 40.5
+(x) S(J) = 1167.7l2 + 714.23l + 4670.9ml + 1167.7.
+Proof. Let f(x, y) = NM(J; x, y) = 6lx5y5 +12lx5y8 +6lx8y8 +12lx8y9 +(9l2 −15l+9+36ml)x9y9.
+Then, we have
+Dx(f(x, y)) = 30lx5y5 + 60lx5y8 + 48lx8y8 + 96lx8y9 + 9(9l2 − 15l + 9 + 36ml)x9y9.
+Dy(f(x, y)) = 30lx5y5 + 96lx5y8 + 48lx8y8 + 108lx8y9 + 9(9l2 − 15l + 9 + 36ml)x9y9.
+D2
+x(f(x, y)) = 150lx5y5 + 300lx5y8 + 384lx8y8 + 768lx8y9 + 81(9l2 − 15l + 9 + 36ml)x9y9.
+D2
+y(f(x, y)) = 150lx5y5 + 768lx5y8 + 384lx8y8 + 972lx8y9 + 81(9l2 − 15l + 9 + 36ml)x9y9.
+DxDy(f(x, y)) = 150lx5y5 + 480lx5y8 + 384lx8y8 + 864lx8y9 + 81(9l2 − 15l + 9 + 36ml)x9y9.
+(Dx + Dy)f(x, y) = 60lx5y5 + 156lx5y8 + 96lx8y8 + 204lx8y9 + 18(9l2 − 15l + 9 + 36ml)x9y9.
+DxDy(Dx + Dy)f(x, y)
+=
+1500lx5y5 + 6240lx5y8 + 6144lx8y8 + 14688lx8y9 + 1458(9l2 − 15l +
+9 + 36ml)x9y9.
+(D2
+x + D2
+y)f(x, y) = 300lx5y5 + 1068lx5y8 + 768lx8y8 + 1740lx8y9 + 162(9l2 − 15l + 9 + 36ml)x9y9.
+Dα
+xDα
+y (f(x, y))
+=
+6l(25)αx5y5 + 12l(40)αx5y8 + 6l(64)αx8y8 + 12l(72)αx8y9 + (81)α
+(9l2 − 15l + 9 + 36ml)x9y9.
+8
+
+SxSy(f(x, y)) = 6l
+25x5y5 + 12l
+40 x5y8 + 6l
+64x8y8 + 12l
+72 x8y9 + (9l2−15l+9+36ml)
+81
+x9y9.
+SyDx + SxDy(f(x, y)) = 12lx5y5 + 267l
+10 x5y8 + 12lx8y8 + 145l
+6 x8y9 + 2(9l2 − 15l + 9 + 36ml)x9y9.
+2SxT(f(x, y)) = 6l
+5 x10 + 24l
+13 x13 + 3l
+4 x16 + 24l
+17 x17 + (9l2−15l+9+36ml)
+9
+x18.
+SxTDxDy(f(x, y)) = 15lx10 + 480l
+13 x13 + 384l
+16 x16 + 864l
+17 x17 + 81(9l2−15l+9+36ml)
+18
+x18.
+S3
+xQ−2TD3
+xD3
+y(f(x, y)) = 93750l
+512 x8+ 768000l
+1331 x11+ 1572864l
+2744
+x14+ 4478976l
+3375
+x15+ 531441(9l2−15l+9+36ml)
+4096
+x16.
+Now, using Table 1 we have
+(i) NM1(J) = (Dx + Dy)f(x, y)|x=y=1 = 162l2 + 246l + 648ml + 162.
+(ii) NM2(J) = (DxDy)f(x, y)|x=y=1 = 729l2 + 663l + 2916ml + 729.
+(iii) NF(J) = (D2
+x + D2
+y)f(x, y)|x=y=1 = 1458l2 + 1446l + 5832ml + 1458.
+(iv) nmM2(J) = (SxSy)f(x, y)|x=y=1 = 0.11l2 + 0.62l + 0.44ml + 0.11.
+(v) NRα(J) = (Dα
+xDα
+y )f(x, y)|x=y=1 = 6l(25α+2(40)α+64α+2(72)α)+81α(9l2−15l+9+36ml).
+(vi) ND3(J) = DxDy(Dx + Dy)f(x, y)|x=y=1 = 13122l2 + 6702l + 52488ml + 13122.
+(vii) ND5(J) = SyDx + SxDy(f(x, y))|x=y=1 = 18l2 + 44.86l + 72ml + 18.
+(viii) NH(J) = 2SxT(f(x, y))|x=y=1 = l2 + 9.69l + 4ml + 1.
+(ix) NI(J) = SxTDxDy(f(x, y))|x=y=1 = 40.5l2 + 59.24l + 162ml + 40.5.
+(x) S(J) = S3
+xQ−2TD3
+xD3
+y(f(x, y))|x=y=1 = 1167.7l2 + 714.23l + 4670.9ml + 1167.7.
+Theorem 3. Let J1 be the second type Y-junction graph of an uncapped symmetrical single-walled
+armchair carbon nanotube. Then
+NM(J1; x, y) = 4lx3y7+4lx5y5+8lx5y8+2lx7y7+4lx7y9+4lx8y8+8lx8y9+(9l2−9l+9+36ml)x9y9.
+Proof. The second type Y-junction graph of an uncapped symmetrical single-walled armchair carbon
+nanotubes has 9l2 +25l+9+36ml edges. Let E(i,j) be the set of all edges with neighborhood degree
+sum of end vertices i, j, i.e., E(i,j) = {uv ∈ E(J1) : dn(u) = i, dn(v) = j}.
+By means of structure analysis of J1, the edge set of J1 can be partitioned into eight sets on the
+basis of neighborhood degree sum of end vertices as follows:
+E(3,7) = {uv ∈ E(J1) : dn(u) = 3, dn(v) = 7}, E(5,5) = {uv ∈ E(J1) : dn(u) = 5, dn(v) = 5},
+E(5,8) = {uv ∈ E(J1) : dn(u) = 5, dn(v) = 8}, E(7,7) = {uv ∈ E(J1) : dn(u) = 7, dn(v) = 7},
+E(7,9) = {uv ∈ E(J1) : dn(u) = 7, dn(v) = 9}, E(8,8) = {uv ∈ E(J1) : dn(u) = 8, dn(v) = 8},
+E(8,9) = {uv ∈ E(J1) : dn(u) = 8, dn(v) = 9}, E(9,9) = {uv ∈ E(J1) : dn(u) = 9, dn(v) = 9},
+and |E(3,7)| = 4l, |E(5,5)| = 4l, |E(5,8)| = 8l, |E(7,7)| = 2l, |E(7,9)| = 4l, |E(8,8)| = 4l, |E(8,9)| = 8l,
+|E(9,9)| = 9l2 − 9l + 9 + 36ml.
+From Equation (1), the NM-polynomial of J1 is obtained as follows:
+NM(J1; x, y)
+=
+�
+i≤j
+|E(i,j)|xiyj
+=
+|E(3,7)|x3y7 + |E(5,5)|x5y5 + |E(5,8)|x5y8 + |E(7,7)|x7y7 + |E(7,9)|x7y9 +
+|E(8,8)|x8y8 + |E(8,9)|x8y9 + |E(9,9)|x9y9
+=
+4lx3y7 + 4lx5y5 + 8lx5y8 + 2lx7y7 + 4lx7y9 + 4lx8y8 + 8lx8y9 +
+(9l2 − 9l + 9 + 36ml)x9y9.
+Theorem 4. Let J1 be the second type Y-junction graph of an uncapped symmetrical single-walled
+armchair carbon nanotube. Then
+9
+
+(i) NM1(J1) = 162l2 + 314l + 648ml + 162
+(ii) NM2(J1) = 729l2 + 957l + 2916ml + 729
+(iii) NF(J1) = 1458l2 + 2074l + 5832ml + 1458
+(iv) nmM2(J1) = 0.11l2 + 0.72l + 0.44ml + 0.11
+(v) NRα(J1)
+=
+2l(2(21)α + 2(25)α + 4(40)α + (49)α + 2(63)α + 2(64)α + 4(72)α) + (81)α(9l2 − 9l + 9
++36ml)
+(vi) ND3(J1) = 13122l2 + 12170l + 52488ml + 13122
+(vii) ND5(J1) = 18l2 + 56.328l + 72ml + 18
+(viii) NH(J1) = l2 + 3.98l + 4ml + 1
+(ix) NI(J1) = 40.5l2 + 75.15l + 162ml + 40.5
+(x) S(J1) = 1167.7l2 + 1178.92l + 4670.9ml + 1167.7.
+Proof. Refer to Theorem 2 for proof.
+Theorem 5. Let J2 be the third type Y-junction graph of an uncapped symmetrical single-walled
+armchair carbon nanotube. Then
+NM(J2; x, y) = 8lx3y7+2lx5y5+4lx5y8+4lx7y7+8lx7y9+2lx8y8+4lx8y9+(9l2−3l+9+36ml)x9y9.
+Proof. The third type Y-junction graph of an uncapped symmetrical single-walled armchair carbon
+nanotubes has 9l2 + 29l + 9 + 36ml number of edges. Let E(i,j) be the set of all edges with neigh-
+borhood degree sum of end vertices i, j, i.e., E(i,j) = {uv ∈ E(J2) : dn(u) = i, dn(v) = j}.
+By means of structure analysis of J2, the edge set of J2 can be partitioned into eight sets on the
+basis of neighborhood degree sum of end vertices as follows:
+E(3,7) = {uv ∈ E(J2) : dn(u) = 3, dn(v) = 7}, E(5,5) = {uv ∈ E(J2) : dn(u) = 5, dn(v) = 5},
+E(5,8) = {uv ∈ E(J)
+2 : dn(u) = 5, dn(v) = 8}, E(7,7) = {uv ∈ E(J2) : dn(u) = 7, dn(v) = 7},
+E(7,9) = {uv ∈ E(J2) : dn(u) = 7, dn(v) = 9}, E(8,8) = {uv ∈ E(J2) : dn(u) = 8, dn(v) = 8},
+E(8,9) = {uv ∈ E(J2) : dn(u) = 8, dn(v) = 9}, E(9,9) = {uv ∈ E(J2) : dn(u) = 9, dn(v) = 9},
+and |E(3,7)| = 8l, |E(5,5)| = 2l, |E(5,8)| = 4l, |E(7,7)| = 4l, |E(7,9)| = 8l, |E(8,8)| = 2l, |E(8,9)| = 4l,
+|E(9,9)| = 9l2 − 3l + 9 + 36ml.
+From Equation (1), the NM-polynomial of J2 is obtained as follows:
+NM(J2; x, y)
+=
+�
+i≤j
+|E(i,j)|xiyj
+=
+|E(3,7)|x3y7 + |E(5,5)|x5y5 + |E(5,8)|x5y8 + |E(7,7)|x7y7 + |E(7,9)|x7y9 +
+|E(8,8)|x8y8 + |E(8,9)|x8y9 + |E(9,9)|x9y9
+=
+8lx3y7 + 2lx5y5 + 4lx5y8 + 4lx7y7 + 8lx7y9 + 2lx8y8 + 4lx8y9 +
+(9l2 − 3l + 9 + 36ml)x9y9.
+Theorem 6. Let J2 be the third type Y-junction graph of an uncapped symmetrical single-walled
+armchair carbon nanotube. Then
+(i) NM1(J2) = 162l2 + 382l + 648ml + 162
+(ii) NM2(J2) = 729l2 + 1251l + 2916ml + 729
+(iii) NF(J2) = 1458l2 + 2478l + 5832ml + 1458
+(iv) nmM2(J2) = 0.11l2 + 0.819l + 0.44ml + 0.11
+10
+
+(v) NRα(J2) = 2l(4(21)α+(25)α+2(40)α+2(49)α+4(63)α+(64)α+2(72)α)+(81)α(9l2−3l+9+36ml)
+(vi) ND3(J2) = 13122l2 + 17638l + 52488ml + 13122
+(vii) ND5(J2) = 18l2 + 65.56l + 72ml + 18
+(viii) NH(J2) = l2 + 4.57l + 4ml + 1
+(ix) NI(J2) = 40.5l2 + 91.048l + 162ml + 40.5
+(x) S(J2) = 1167.7l2 + 1643.61l + 4670.9ml + 1167.7.
+Proof. Refer to Theorem 2 for proof.
+Theorem 7. Let J3 be the fourth type Y-junction graph of an uncapped symmetrical single-walled
+armchair carbon nanotube. Then
+NM(J3; x, y) = 12lx3y7 + 6lx7y7 + 12lx7y9 + (9l2 + 3l + 9 + 36ml)x9y9.
+Proof. The fourth type Y-junction graph of an uncapped symmetrical single-walled armchair car-
+bon nanotube has 9l2 + 33l + 9 + 36ml number of edges. Let E(i,j) be the set of all edges with
+neighborhood degree sum of end vertices i, j, i.e., E(i,j) = {uv ∈ E(J3) : dn(u) = i, dn(v) = j}.
+By means of structure analysis of J3, the edge set of J3 can be partitioned into four sets on the basis
+of neighborhood degree sum of end vertices as follows:
+E(3,7) = {uv ∈ E(J3) : dn(u) = 3, dn(v) = 7}, E(7,7) = {uv ∈ E(J3) : dn(u) = 7, dn(v) = 7},
+E(7,9) = {uv ∈ E(J3) : dn(u) = 7, dn(v) = 9}, E(9,9) = {uv ∈ E(J3) : dn(u) = 9, dn(v) = 9}, and
+|E(3,7)| = 12l, |E(7,7)| = 6l, |E(7,9)| = 12l, |E(9,9)| = 9l2 + 3l + 9 + 36ml.
+From Equation (1), the NM-polynomial of J3 is obtained as follows:
+NM(J3; x, y)
+=
+�
+i≤j
+|E(i,j)|xiyj
+=
+|E(3,7)|x3y7 + |E(7,7)|x7y7 + |E(7,9)|x7y9 + |E(9,9)|x9y9
+=
+12lx3y7 + 6lx7y7 + 12lx7y9 + (9l2 + 3l + 9 + 36ml)x9y9.
+Theorem 8. Let J3 be the fourth type Y-junction graph of an uncapped symmetrical single-walled
+armchair carbon nanotube. Then
+(i) NM1(J3) = 162l2 + 450l + 648ml + 162
+(ii) NM2(J3) = 729l2 + 1545l + 2916ml + 729
+(iii) NF(J3) = 1458l2 + 3330l + 5832ml + 1458
+(iv) nmM2(J3) = 0.11l2 + 0.92l + 0.44ml + 0.11
+(v) NRα(J3) = 6l(2(21)α + (49)α + 2(63)α) + (81)α(9l2 + 3l + 9 + 36ml)
+(vi) ND3(J3) = 13122l2 + 23106l + 52488ml + 13122
+(vii) ND5(J3) = 18l2 + 75.90l + 72ml + 18
+(viii) NH(J3) = l2 + 5.090l + 4ml + 1
+(ix) NI(J3) = 40.5l2 + 106.95l + 162ml + 40.5
+(x) S(J3) = 1167.7l2 + 2085.95l + 4670.9ml + 1167.7.
+Proof. Refer to Theorem 2 for proof.
+11
+
+6
+Graph Index-Entropies of Y-Junction Graphs
+In this section, we compute the index-entropy of carbon nanotube Y-junctions in terms of neigh-
+borhood degree sum-based topological indices. We first compute index-entropies of the Y-junction
+graph J whose edge partition is given in Table 2.
+• Third-version of Zagreb index-entropy of J
+From part (i) of Theorem 2, we have
+NM1(J) = 162l2 + 246l + 648ml + 162.
+(19)
+Now, from Equation (4), the third-version of Zagreb index-entropy of J is
+Hβ1(J) = log(NM1(J)) −
+1
+NM1(J)
+�
+e∈E(J)
+β1(e)logβ1(e).
+(20)
+Using Table 2 and Equation (19) in Equation (20), we get the required third-version of Zagreb
+index-entropy of J as follows:
+Hβ1(J)
+=
+log(NM1(J)) −
+1
+NM1(J)
+�
+e∈E(J)
+β1(e)logβ1(e)
+=
+log(162l2 + 246l + 648ml + 162) −
+1
+162l2 + 246l + 648ml + 162
+�
+6l(10)(log10) +
+12l(13)(log13) + 6l(16)(log16) + 12l(17)(log17) + (9l2 − 15l + 36ml + 9)(18)(log18)
+�
+=
+log(162l2 + 246l + 648ml + 162) −
+1
+162l2 + 246l + 648ml + 162
+�
+60l(log10) +
+156l(log13) + 96l(log16) + 204l(log17) + (162l2 − 270l + 648ml + 162)(log18)
+�
+=
+log(162l2 + 246l + 648ml + 162) −
+1
+162l2 + 246l + 648ml + 162
+�
+60l(1) + 156l(1.1139433523)
++96l(1.2041199827) + 204l(1.2304489214) + (162l2 − 270l + 648ml + 162)(1.2552725051)
+�
+≈ log(162l2 + 246l + 648ml + 162) − 202.5l2 + 261.78l + 810ml + 202.5
+162l2 + 246l + 648ml + 162
+.
+• Neighborhood second Zagreb index-entropy of J
+From part (ii) of Theorem 2, we have
+NM2(J) = 729l2 + 663l + 2916ml + 729.
+(21)
+By using the values given in Table 2 and Equation (21) in Equation (6), we get the required neigh-
+borhood second Zagreb index-entropy of J as follows:
+Hβ2(J)
+=
+log(NM2(J)) −
+1
+NM2(J)
+�
+e∈E(J)
+β2(e)logβ2(e)
+=
+log(729l2 + 663l + 2916ml + 729) −
+1
+729l2 + 663l + 2916ml + 729
+�
+6l(25)(log25) +
+12l(40)(log40) + 6l(64)(log64) + 12l(72)(log72) + (9l2 − 15l + 36ml + 9)(81)(log81)
+�
+≈ log(729l2 + 663l + 2916ml + 729) − 1391.22l2 + 958.27l + 5564.88ml + 1391.22
+729l2 + 663l + 2916ml + 729
+.
+Similarly, we compute the remaning index-entropies of J. Table 3 shows some calculated graph
+index-entropies of J.
+In this way, the topological index-based entropies for Y-junction graphs J1, J2, and J3 are
+calculated.
+The index-based entropies of J1 J2, and J3 are given in Tables 4, 5, and 6.
+12
+
+Table 3: Index-entropies of J
+Entropy
+Values of entropies
+Hβ3(J)
+log(1458l2 + 1446l + 5832ml + 1458) − 3221.62l2+2601.62l+12885.84ml+3221.46
+1458l2+1446l+5832ml+1458
+Hβ4(J)
+log(0.11l2 + 0.62l + 0.44ml + 0.11) + 0.207l2+0.92l+0.828ml+0.207
+0.11l2+0.62l+0.44ml+0.11
+Hβ5(J)
+log(13122l2 + 12170l + 52488ml + 13122) − 41514.75l2+15202.13l+166059.31ml+41514.75
+13122l2+12170l+52488ml+13122
+Hβ6(J)
+log(18l2 + 44.86l + 72ml + 18) − 5.41l2+14.81l+21.67ml+5.41
+18l2+44.86l+72ml+18
+Hβ7(J)
+log(l2 + 9.69l + 4ml + 1) + 0.95l2+2.72l+3.81ml+0.95
+l2+9.69l+4ml+1
+Hβ8(J)
+log(40.5l2 + 59.24l + 162ml + 40.5) − 26.45l2+26.18l+105.82ml+26.45
+40.5l2+59.24l+162ml+40.5
+Table 4: Index-entropies of J1
+Entropy
+Values of entropies
+Hβ1(J1)
+log(162l2 + 314l + 648ml + 162) − 203.31l2+346.09l+813.24ml+203.31
+162l2+314l+648ml+162
+Hβ2(J1)
+log(729l2 + 957l + 2916ml + 729) − 1391.22l2+1523.54l+5564.88ml+1391.22
+729l2+957l+2916ml+729
+Hβ3(J1)
+log(1458l2 + 2074l + 5832ml + 1458) − 3221.46l2+3991l+12885.84ml+3221.46
+1458l2+2074l+5832ml+1458
+Hβ4(J1)
+log(0.11l2 + 0.72l + 0.44ml + 0.11) + 0.207l2+1.065l+0.897ml+0.207
+0.11l2+0.72l+0.44ml+0.11
+Hβ5(J1)
+log(13122l2 + 12170l + 52488ml + 13122) − 41514.75l2+32699.53l+166059ml+41514.75
+13122l2+12170l+52488ml+13122
+Hβ6(J1)
+log(18l2 + 56.328l + 72ml + 18) − 5.41l2+19.09l+21.67ml+5.41
+18l2+56.328l+72ml+18
+Hβ7(J1)
+log(l2 + 3.98l + 4ml + 1) + 0.9l2+3.21l+3.6ml+0.9
+l2+3.98l+4ml+1
+Hβ8(J1)
+log(40.5l2 + 75.15l + 162ml + 40.5) − 26.37l2+36.84l+105.48ml+26.37
+40.5l2+75.15l+162ml+40.5
+Table 5: Index-entropies of J2
+Entropy
+Values of entropies
+Hβ1(J2)
+log(162l2 + 382l + 648ml + 162) − 203.31l2+430.65l+813.24ml+203.31
+162l2+382l+648ml+162
+Hβ2(J2)
+log(729l2 + 1251l + 2916ml + 729) − 1391.22l2+2088.77l+5564.88ml+1391.22
+729l2+1251l+2916ml+729
+Hβ3(J2)
+log(1458l2 + 2478l + 5832ml + 1458) − 3221.46l2+5380.37l+12885.84ml+3221.46
+1458l2+2478l+5832ml+1458
+Hβ4(J2)
+log(0.11l2 + 0.819l + 0.44ml + 0.11) + 0.099l2+1.007l+0.396ml+0.099
+0.11l2+0.819l+0.44ml+0.11
+Hβ5(J2)
+log(13122l2 + 17638l + 52488ml + 13122) − 41514.75l2+50196.95l+166059ml+50196.95
+13122l2+17638l+52488ml+13122
+Hβ6(J2)
+log(18l2 + 65.56l + 72ml + 18) − 5.41l2+23.44l+21.67ml+5.41
+18l2+65.56l+72ml+18
+Hβ7(J2)
+log(l2 + 4.57l + 4ml + 1) + 0.9l2+3.61l+3.6ml+0.9
+l2+4.57l+4ml+1
+Hβ8(J2)
+log(40.5l2 + 91.048l + 162ml + 40.5) − 26.37l2+46.387l+105.48ml+26.37
+40.5l2+91.048l+162ml+40.5
+13
+
+Table 6: Index-entropies of J3
+Entropy
+Values of entropies
+Hβ1(J3)
+log(162l2 + 450l + 648ml + 162) − 203.31l2+515.23l+813.24ml+203.31
+162l2+450l+648ml+162
+Hβ2(J3)
+log(729l2 + 1545l + 2916ml + 729) − 1391.22l2+2654.14l+5564.88ml+1391.22
+729l2+1545l+2916ml+729
+Hβ3(J3)
+log(1458l2 + 3330l + 5832ml + 1458) − 3221.46l2+6769.75l+12885.84ml+3221.46
+1458l2+3330l+5832ml+1458
+Hβ4(J3)
+log(0.11l2 + 0.92l + 0.44ml + 0.11) + 0.18l2+1.35l+0.72ml+0.18
+0.11l2+0.92l+0.44ml+0.11
+Hβ5(J3)
+log(13122l2 + 23106l + 52488ml + 13122) − 41514.75l2+67694.4l+166059ml+41514
+13122l2+23106l+52488ml+13122
+Hβ6(J3)
+log(18l2 + 75.90l + 72ml + 18) − 5.41l2+27.79l+21.67ml+5.41
+18l2+75.90l+72ml+18
+Hβ7(J3)
+log(l2 + 5.090l + 4ml + 1) + 0.9l2+4.04l+3.6ml+0.9
+l2+5.090l+4ml+1
+Hβ8(J3)
+log(40.5l2 + 106.95l + 162ml + 40.5) − 26.37l2+56.44l+105.48ml+26.37
+40.5l2+106.95l+162ml+40.5
+7
+Numerical Results and Discussions
+The numerical values of topological indices and graph index-entropies of Y-junction graphs are com-
+puted in this section for some values of l and m.
+In addition, we plot line and bar graphs for
+comparison of the obtained results. Here, we use the logarithm of the base 10 for calculations.
+The numerical values of topological indices for Y-junction graph J are given in Table 7. The
+logarithmic values of Table 7 are plotted in Figure 2. From the vertical axis of Figure 2, we can
+conclude that for Y-junction graph J, the topological indices have the following order:
+nmM2 ≤
+NR−1/2 ≤ NH ≤ ND5 ≤ NI ≤ NM1 ≤ NM2 ≤ S ≤ NF ≤ ND3. The third NDe index has
+the most dominating nature compared to other topological indices, whereas neighborhood second
+modified Zagreb index grew slowly.
+Table 7: Numerical values of topological indices for Y-junction graph J
+[l, m]
+NM1(J)
+NM2(J)
+NF (J)
+NmM2(J)
+NR− 1
+2
+(J)
+ND3(J)
+ND5(J)
+NH(J)
+NI(J)
+S(J)
+[2,2]
+3894
+16635
+33510
+3.55
+20.7436
+288966
+467.72
+40.38
+968.92
+25950.56
+[3,3]
+8190
+35523
+71406
+6.92
+45.27693
+623718
+962.58
+75.07
+2040.63
+55857.79
+[4,4]
+14106
+61701
+123882
+11.39
+79.81026
+1089690
+1637.44
+119.76
+3517.34
+97442.22
+[5,5]
+21642
+95169
+190938
+16.96
+124.3436
+1686882
+2492.3
+174.45
+5399.05
+150703.9
+[6,6]
+30798
+135927
+272574
+23.63
+178.8769
+2415294
+3527.16
+239.14
+7685.76
+215642.7
+[7,7]
+41574
+183975
+368790
+31.4
+243.4103
+3274926
+4742.02
+313.83
+10377.47
+292258.7
+[8,8]
+53970
+239313
+479586
+40.27
+317.9436
+4265778
+6136.88
+398.52
+13474.18
+380551.9
+[9.9]
+67986
+301941
+604962
+50.24
+440.4769
+5387850
+7711.74
+493.21
+16975.89
+480522.4
+[10,10]
+83622
+371859
+744918
+61.31
+497.0103
+6641142
+9466.6
+597.9
+20882.6
+592170
+Figure 2: Graphical comparison among topological indices of Y-junction graph J
+14
+
+indices
+1000000
+NM,(J)
+100000
+NF(J)
+nmr
+M.(J)
+NR
+10000
+D
+(J
+1000
+NH(J)
+NI(J)
+S(J)
+100
+10
+[4,4]
+[5,5]
+[6,6]
+[7,7]
+[2,2] 
+[3,3]
+[8,8] 
+[9,9] [10,10]
+[1,m]Table 8 shows some numerical values of topological indices for Y-junction graph J1. The logarith-
+mic values of these topological indices are plotted in Figure 3. From Figure 3, we can conclude that
+the topological indices for Y-junction graph J1 have the following order: nmM2 ≤ NH ≤ NR−1/2 ≤
+ND5 ≤ NI ≤ NM1 ≤ NM2 ≤ S ≤ NF ≤ ND3. Also, we see that the logarithemic values of
+NR−1/2 and NH for J1 are almost same.
+Table 8: Numerical values of topological indices for Y-junction graph J1
+[l, m]
+NM1(J1)
+NM2(J1)
+NF (J1)
+nmM2(J1)
+NR− 1
+2
+(J1)
+ND3(J1)
+ND5(J1)
+NH(J1)
+NI(J1)
+S(J1)
+[2,2]
+4030
+17223
+35166
+3.75
+29.34052
+299902
+490.656
+28.96
+1000.8
+26879.94
+[3,3]
+8394
+36405
+74190
+7.22
+58.51078
+640122
+996.984
+57.94
+2088.45
+57251.86
+[4,4]
+14378
+62877
+127994
+11.79
+97.68103
+1111562
+1683.312
+96.92
+3581.1
+99300.98
+[5,5]
+21982
+96639
+196578
+17.46
+146.8513
+1714222
+2549.64
+145.9
+5478.75
+153027.3
+[6,6]
+31206
+137691
+279942
+24.23
+206.0216
+2448102
+3595.968
+204.88
+7781.4
+218430.8
+[7,7]
+42050
+186033
+378086
+32.1
+275.1918
+3313202
+4822.296
+273.86
+10489.05
+295511.5
+[8,8]
+54514
+241665
+491010
+41.07
+354.3621
+4309522
+6228.624
+352.84
+13601.7
+384269.5
+[9.9]
+68598
+304587
+618714
+51.14
+443.5323
+5437062
+7814.952
+441.82
+17119.35
+484704.6
+[10,10]
+84302
+374799
+761198
+62.31
+542.7026
+6695822
+9581.28
+540.8
+21042
+596816.9
+Figure 3: Graphical comparison among topological indices of Y-junction graph J1
+Table 9 shows some calculated values of topological indices for Y-junction graph J2. The log-
+arithmic values of these indices are plotted in Figure 4. The vertical axis of Figure 4 shows the
+comparison clearly. Figure 4 shows that the logarithmic values of ND3 are extremely high when
+compared to other topological indices of J2. From Figure 4, we see that the graph of NR−1/2 and
+NH are almost coincide.
+Table 9: Numerical values of topological indices for Y-junction graph J2
+[l, m]
+NM1(J2)
+NM2(J2)
+NF (J2)
+nmM2(J2)
+NR− 1
+2
+(J2)
+ND3(J2)
+ND5(J2)
+NH(J2)
+NI(J2)
+S(J2)
+[2,2]
+4166
+17811
+35574
+3.948
+30.49121
+310838
+509.12
+30.14
+1032.596
+27809.32
+[3,3]
+8598
+37287
+74502
+7.517
+60.23681
+656526
+1024.68
+59.71
+2136.144
+58645.93
+[4,4]
+14650
+64053
+128010
+12.186
+99.98241
+1133434
+1720.24
+99.28
+3644.692
+101159.7
+[5,5]
+22322
+98109
+196098
+17.955
+149.728
+1741562
+2595.8
+148.85
+5558.24
+155350.8
+[6,6]
+31614
+139455
+278766
+24.824
+209.2192
+2480910
+3651.36
+208.42
+7876.788
+221219
+[7,7]
+42526
+188091
+376014
+32.793
+279.2192
+3351478
+4886.92
+277.99
+10600.34
+298764.4
+[8,8]
+55058
+244017
+487842
+41.862
+358.9648
+4353266
+6302.48
+357.56
+13728.88
+387987
+[9,9]
+69210
+307233
+614250
+52.031
+448.7104
+5486274
+7898.48
+447.13
+17262.43
+488886.8
+[10,10]
+84982
+377739
+755238
+63.3
+548.456
+6750502
+9673.6
+546.7
+21200.98
+601463.8
+15
+
+indices
+1000000
+-NM,(J,)
+100000
+NM,(J,)
+NF(J))
+10000
+NR
+(J1)
+ND,(J,)
+1000
+ND,(J)
+NH(J,)
+100
+NI(J)
+10
+[2,2] 
+[3,3]
+[4,4]
+[5,5]
+[6,6]
+[7,7]
+[8,8] 
+[9,9][10,10]
+[1,m]Figure 4: Graphical comparison among topological indices of Y-junction J2
+Table 10 shows some numerical values of topological indices of Y-junction J3. Figure 5 depicts
+the graphical comparison of these indices. Table 10 and Figure 5 show that the values of topological
+indices strictly increase as the values of l and m increases.
+From Tables 7, 8, 9, and 10, we see that as the values of l and m in Y-junction graphs increases, the
+corresponding values of topological indices grew very fastly.
+Table 10: Numerical values of topological indices of Y-junction graph J3
+[l, m]
+NM1(J3)
+NM2(J3)
+NF (J3)
+NmM2(J3)
+NR− 1
+2
+(J3)
+ND3(J3)
+ND5(J3)
+NH(J3)
+NI(J3)
+S(J3)
+[2,2]
+4302
+18399
+37278
+4.15
+31.6419
+321774
+529.8
+31.18
+1064.4
+28694
+[3,3]
+8802
+38169
+77058
+7.82
+61.9628
+672930
+1055.7
+61.27
+2183.85
+59973
+[4,4]
+14922
+65229
+131418
+12.59
+102.284
+1155306
+1761.6
+101.36
+3708.3
+102929
+[5,5]
+22662
+99579
+200358
+18.46
+152.605
+1768902
+2647.5
+151.45
+5637.75
+157562
+[6,6]
+32022
+141219
+283878
+25.43
+212.926
+2513718
+3713.4
+211.54
+7972.2
+223873
+[7,7]
+43002
+190149
+381978
+33.5
+283.247
+3389754
+4959.3
+281.63
+10711.7
+301861
+[8,8]
+55602
+246369
+494658
+42.67
+363.568
+4397010
+6385.2
+361.72
+13856.1
+391526
+[9,9]
+69822
+309879
+621918
+52.94
+453.889
+5535486
+7991.1
+451.81
+17405.6
+492868
+[10,10]
+85662
+380679
+763758
+64.31
+554.209
+6805182
+9777
+551.9
+21360
+605887
+Figure 5: Graphical comparison among topological indices of Y-junction graph J3
+A few values of graph index-entropies of Y-junction graph J are listed in Table 11 and illustrated
+in Figure 6.
+From Figure 6, we see that entropy measures of Hβ1, Hβ2, Hβ3, and Hβ8 almost
+16
+
+indices
+1000000
+100000
+NF(J.
+10000
+NR
+1000
+ND,(J)
+ND,(J2)
+100
+- NH(J,)
+- NI(J2)
+10
+[2,2] 
+[3,3]
+[4,4]
+[5,5]
+[6,6]
+[7,7]
+[8,8] 
+[9,9][10,10]
+[1,m]indices
+1000000
+100000
+NM(J.
+NM,(J3)
+NF(J3)
+10000
+"M,(J3)
+NR.1/2(J3)
+ND,(J3)
+1000
+ND,(J3)
+NH(J3)
+NI(J3)
+100
+S(J3)
+10
+[2,2] 
+[4,4]
+[5,5]
+[6,6]
+[7,7]
+[3,3]
+[8,8] 
+[9,9][10,10]
+[1,m]coincide.
+Table 11: Numerical values of index-entropies of J
+[l, m]
+Hβ1 (J)
+Hβ2 (J)
+Hβ3 (J)
+Hβ4 (J)
+Hβ5 (J)
+Hβ6 (J)
+Hβ7 (J)
+Hβ8 (J)
+[2,2]
+2.363878
+2.349537
+2.351098
+2.293045
+2.469266
+1.849911
+2.235934
+2.358964
+[3,3]
+2.680031
+2.668041
+2.669162
+2.614962
+2.751708
+2.162725
+2.567487
+2.674998
+[4,4]
+2.912369
+2.901799
+2.902659
+2.851695
+2.966026
+2.393078
+2.813031
+2.907277
+[5,5]
+3.09586
+3.086229
+3.086919
+3.038506
+3.138274
+2.575276
+3.00722
+3.090734
+[6,6]
+3.24743
+3.238462
+3.239031
+3.192634
+3.28218
+2.725919
+3.167437
+3.242282
+[7,7]
+3.37652
+3.368045
+3.368525
+3.323745
+3.40572
+2.85433
+3.303596
+3.371357
+[8,8]
+3.488993
+3.480834
+3.481247
+3.437785
+3.51397
+2.966216
+3.421864
+3.483755
+[9.9]
+3.588472
+3.580678
+3.581037
+3.538669
+3.610178
+3.065343
+3.526328
+3.583289
+[10,10]
+3.677788
+3.670239
+3.670554
+3.629107
+3.696846
+3.154321
+3.61983
+3.672599
+Figure 6: Graphical comparison among index-entropies of J
+The values of index-entropy of Y-junction graph J1 is listed in Table 12 and illustrated in Figure
+7. From Table 12 and Figure 7, we find that measures of graph index-entropies Hβ1, Hβ2, Hβ3, Hβ5,
+Hβ6, and Hβ1 are almost same.
+Table 12: Numerical values of index-entropies of J1
+[l, m]
+Hβ1 (J1)
+Hβ2 (J1)
+Hβ3 (J1)
+Hβ4 (J1)
+Hβ5 (J1)
+Hβ6 (J1)
+Hβ7 (J1)
+Hβ8 (J1)
+[2,2]
+2.374116
+2.362875
+2.365677
+2.37483
+2.351939
+2.381171
+2.336108
+2.373399
+[3,3]
+2.686117
+2.677718
+2.679751
+2.705906
+2.66972
+2.691361
+2.643717
+2.686081
+[4,4]
+2.916047
+2.909359
+2.91094
+2.948613
+2.903076
+2.92019
+2.871061
+2.916411
+[5,5]
+3.097981
+3.092424
+3.093709
+3.139639
+3.087253
+3.101393
+3.051307
+3.098605
+[6,6]
+3.248463
+3.243705
+3.244784
+3.2969
+3.239312
+3.251355
+3.200605
+3.24927
+[7,7]
+3.376753
+3.372588
+3.373514
+3.430435
+3.368768
+3.379258
+3.23802
+3.377694
+[8,8]
+3.488549
+3.484842
+3.485651
+3.546395
+3.481462
+3.490754
+3.439143
+3.489594
+[9.9]
+3.587606
+3.584263
+3.584979
+3.648847
+3.58132
+3.589572
+3.537668
+3.588733
+[10,10]
+3.676529
+3.673482
+3.674123
+3.740586
+3.6707383
+3.6783
+3.626158
+3.677722
+17
+
+3.8
+Hβ,(J)
+3.6
+Hβ,(J)
+Hβ,(J)
+3.4
+Hβ(J)
+3.2
+Hβ,(J)
+Hβ,(J)
+3.0
+Hβ, (J)
+Index-entropies
+2.8
+2.6
+2.4
+2.2
+2.0
+1.8
+[2,2]  [3,3]  [4,4]
+[5,5]
+[6,6]
+[7,7]
+[8,8]
+[9,9] [10,10]
+[1,m]Figure 7: Graphical comparison among index-entropies of J1
+Table 13 depicts some graph index-entropies of Y-junction graph J2. The graphical comparison
+of index-entropies of Y-junction graph J2 is shown in Figure 8. From Figure 8, we see that graph
+index-entropies of J2 increases as the values of l and m increases.
+Table 13: Numerical values of index-entropies of J2
+[l, m]
+Hβ1 (J2)
+Hβ2 (J2)
+Hβ3 (J2)
+Hβ4 (J2)
+Hβ5 (J2)
+Hβ6 (J2)
+Hβ7 (J2)
+Hβ8 (J2)
+[2,2]
+2.388128
+2.375827
+2.346955
+1.633105
+2.336917
+2.391354
+2.345766
+2.35432
+[3,3]
+2.696411
+2.687189
+2.666479
+1.88376
+2.665889
+2.698832
+2.650775
+2.672367
+[4,4]
+2.924151
+2.916793
+2.9007
+2.074455
+2.902766
+2.926068
+2.876596
+2.905747
+[5,5]
+3.104655
+3.098533
+3.085384
+2.229346
+3.088295
+3.106231
+3.055853
+3.089892
+[6,6]
+3.254134
+3.248888
+3.237774
+2.360107
+3.240916
+3.255465
+3.204459
+3.241908
+[7,7]
+3.381681
+3.377087
+3.367462
+2.473394
+3.370602
+3.3882828
+3.331363
+3.371323
+[8,8]
+3.492905
+3.488816
+3.480327
+2.573399
+3.48337
+3.49391
+3.442095
+3.483978
+[9.9]
+3.59151
+3.587821
+3.580028
+2.662948
+3.583139
+3.592399
+3.540309
+3.583714
+[10,10]
+3.680065
+3.676703
+3.659833
+2.744042
+3.672602
+3.680861
+3.628548
+3.673185
+Figure 8: Graphical comparison among index-entropies of J2
+In Table 14, we calculate some graph index-entropies of Y-junction graph J3. Figure 9 shows
+the graphical comparison among index-entropies of J3. From Table 14 and Figure 9, we see that
+index entropies Hβ1, Hβ2, Hβ3, Hβ6, and Hβ8 of J3 are almost same. Also, Tables 11, 12, 13, and 14
+shows that graph index-entropies of Y-junction graph increases as the values of l and m increases.
+18
+
+4.0
+Hβ(J,)
+Hβ,(J,)
+3.5
+Hβ,(J,)
+Hβ(J,)
+Hβ,(J)
+3.0
+Hβ,(J,)
+Hβ,(J,)
+Hβ(J,)
+2.5
+Index-entropies
+2.0
+1.5
+1.0
+0.5
+0.0
+[3,3]
+[2,2]
+[4,4]
+[5,5]
+[7,7]
+[6,6]
+[8,8]
+[9,9]
+[10,10]
+[1,m] Hβ,(J2)
+Hβ,(J2)
+3.5
+Hβ,(J2)
+I Hβ,(J2)
+I Hβ,(J2)
+3.0
+Hβ,(J2)
+Hβ,(J2)
+Hβ,(J2)
+2.5
+Index-entropies
+2.0
+1.5
+1.0
+0.5
+0.0
+[3,3]
+[2,2]
+[4,4]
+[5,5]
+[7,7]
+[6,6]
+[8,8]
+[9,9]
+[10,10]
+[1,m] Table 14: Numerical values of index-entropies of J3
+[l, m]
+Hβ1 (J3)
+Hβ2 (J3)
+Hβ3 (J3)
+Hβ4 (J3)
+Hβ5 (J3)
+Hβ6 (J3)
+Hβ7 (J3)
+Hβ8 (J3)
+[2,2]
+2.401692
+2.388393
+2.393488
+2.179494
+1.755856
+2.404539
+2.359174
+2.40079
+[3,3]
+2.706459
+2.696449
+2.7002
+2.469933
+2.242514
+2.708584
+2.660759
+2.70624
+[4,4]
+2.932101
+2.924095
+2.927043
+2.687
+2.571439
+2.933776
+2.884517
+2.932298
+[5,5]
+3.11224
+3.104551
+3.106972
+2.86049
+2.816575
+3.112596
+3.062409
+3.111698
+[6,6]
+3.259727
+3.254003
+3.256051
+3.005032
+3.010781
+3.260882
+3.210048
+3.260399
+[7,7]
+3.38655
+3.381534
+3.383306
+3.128925
+3.171078
+3.387542
+3.336232
+3.387369
+[8,8]
+3.497215
+3.492748
+3.494307
+3.237341
+3.307302
+3.498082
+3.446407
+3.498149
+[9.9]
+3.595375
+3.591345
+3.592735
+3.33372
+3.425608
+3.596141
+3.544179
+3.5964
+[10,10]
+3.683569
+3.679896
+3.681147
+3.420469
+3.530087
+3.684252
+3.632058
+3.684668
+Figure 9: Graphical comparison among index-entropies of J3
+8
+Conclusion and Future work
+In this study, the general expression of NM-polynomial for carbon nanotube Y-junction graphs is
+derived. Also, various neighborhood degree sum-based topological indices are retrieved from the
+expression of these polynomials.
+In addition, eight graph entropies in terms of these topologi-
+cal indices have been defined and calculated for Y-junction graphs. Furthermore, some numerical
+values of topological indices and index-entropies of Y-junction graphs are plotted for comparison.
+Since topological indices based on the degree of vertices has a significant ability to predict various
+physicochemical properties and biological activities of the chemical molecule. Therefore, the study’s
+findings will be a viable option for predicting various physicochemical properties and understanding
+the structural problems of carbon nanotube Y-junctions.
+We mention some possible directions for future research, including multiplicative topological
+indices, graph index-entropies, regression models between the index-entropies and the topological
+indices, metric and edge metric dimension, etc., to predict thermochemical data, physicochemical
+properties, and structural information of carbon nanotube Y-junctions.
+Data Availability
+No data was used to support the findings of this study.
+Conflicts of Interest
+There are no conflicts of interest declared by the authors.
+Funding Statement
+The authors received no specific funding for this study.
+19
+
+Hβ,(J3)
+Hβ,(J3)
+Hβ,(J3)
+3.5
+Hβ,(J3)
+Hβ,(J3)
+Hβ,(Js)
+3.0
+Hβ,(J3)
+Hβ,(J3)
+2.5
+Index-entropies
+2.0
+1.0
+0.5
+0.0
+[3,3]
+[2,2]
+[4,4]
+[5,5]
+17,7]
+[6,6]
+[8,8]
+[9,9]
+[10,10]
+[1,m]Author’s Contribution Statement
+The final draft was written by Sohan Lal and Vijay Kumar Bhat.
+Figures and Tables are
+prepared by Sohan Lal and Sahil Sharma. All authors reviewed and edited the final draft.
+References
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+

ANFLT4oBgHgl3EQfwzCR/content/tmp_files/2301.12165v1.pdf.txt

@@ -0,0 +1,705 @@
+1
+Dynamic Point Cloud Geometry Compression
+Using Multiscale Inter Conditional Coding
+Jianqiang Wang, Dandan Ding, Hao Chen, and Zhan Ma
+Abstract—This work extends the Multiscale Sparse Repre-
+sentation (MSR) framework developed for static Point Cloud
+Geometry Compression (PCGC) to support the dynamic PCGC
+through the use of multiscale inter conditional coding. To this
+end, the reconstruction of the preceding Point Cloud Geometry
+(PCG) frame is progressively downscaled to generate multi-
+scale temporal priors which are then scale-wise transferred
+and integrated with lower-scale spatial priors from the same
+frame to form the contextual information to improve occupancy
+probability approximation when processing the current PCG
+frame from one scale to another. Following the Common Test
+Conditions (CTC) defined in the standardization committee, the
+proposed method presents State-Of-The-Art (SOTA) compression
+performance, yielding 78% lossy BD-Rate gain to the latest
+standard-compliant V-PCC and 45% lossless bitrate reduction
+to the latest G-PCC. Even for recently-emerged learning-based
+solutions, our method still shows significant performance gains.
+Index Terms—Dynamic point cloud geometry, Multiscale tem-
+poral prior, Inter conditional coding.
+I. INTRODUCTION
+Dynamic point clouds are of great importance for applica-
+tions like holographic communication, autonomous machinery,
+etc., for which the efficient compression of dynamic Point
+Cloud Geometry (PCG) plays a vital role in service provision-
+ing. In addition to rules-based Point Cloud Geometry Com-
+pression (PCGC) technologies standardized by the ISO/IEC
+MPEG (Moving Picture Experts Group), e.g., Video-based
+PCC (V-PCC) and Geometry-based PCC (G-PCC) [3], [4], [5],
+learning-based PCGC methods have been extensively investi-
+gated in the past few years, greatly improving the performance
+with very encouraging prospects [6]. Among those learning-
+based solutions, multiscale sparse representation (MSR) [2],
+[1], [7], [8] has improved the performance unprecedentedly by
+effectively exploiting cross-scale and same-scale correlations
+in the same frame of a static PCG for compact representation.
+The compression of a static PCG frame independently can also
+be referred to as the intra coding of the PCG.
+This work extends the MSR framework originally developed
+for static PCGs to compress the dynamic PCGs [2], [1]. In this
+regard, we suggest the inclusion of multiscale temporal priors
+for inter conditional coding. As in Fig. 1, for a previously-
+reconstructed PCG frame (e.g., PCG t − 1), we progressively
+downsample it and extract scale-wise hierarchical features
+which are then transferred as additional temporal priors to help
+the compression of the same-scale PCG tensor of the current
+frame (e.g., PCG t). To this end, we basically concatenate
+J. Wang, H. Chen and Z. Ma are with Nanjing University, China; D. Ding
+is with Hangzhou Normal University, China.
+the same-scale temporal priors from the inter reference and
+lower-scale spatial priors from the same intra frame to form
+the contextual information for better conditional occupancy
+probability approximation in compression. Such an inter con-
+ditional coding scheme for dynamic PCGC is implemented on
+top of the SparsePCGC [1] originally developed for the static
+PCGC, to quantitatively evaluate its efficiency. Experimental
+results demonstrate the leading performance of our method
+when compared with existing methods (either rules-based
+or learning-based ones) in both lossy and lossless modes,
+following the Common Test Conditions (CTC) used in the
+MPEG standardization committee [9].
+II. RELATED WORK
+In addition to existing G-PCC and V-PCC standards and
+other rules-based PCC methods in [10], [11], [12], [13], [14],
+[15], an excessive number of learning-based PCC solutions
+have emerged in the past years. Therefore the ISO/IEC MPEG
+3D graphics coding group initiated the Artificial Intelligence-
+based Point Cloud Compression (AI-PCC) to investigate po-
+tential technologies for better compression of point clouds.
+Static PCGC. Recently, major endeavors have been paid
+to study the compression of a static PCG [6], a.k.a. Static
+PCGC, yielding voxel-based [16], [17], point-based [18],
+octree-based [19], and sparse tensor-based approaches [7],
+[2], [1]. Among them, sparse tensor-based methods not only
+attain the leading performance but also have low complexity.
+The first representative work is the PCGCv2 [2] where a
+static PCG tensor is hierarchically downsampled and lossily
+compressed using a Sparse Convolutional Neural Network
+(SparseCNN) based autoencoder. Later, the SparsePCGC [1]
+improves the PCGCv2 greatly under a unified MSR framework
+to support both lossless and lossy compression of various point
+clouds by extensively exploiting cross-scale and same-scale
+correlations for better contextual modeling using SparseCNN-
+based Occupancy Probability Approximation (SOPA) models.
+More details regarding the MSR and SOPA model can be
+found in [1].
+Dynamic
+PCGC.
+On
+top
+of
+the
+PCGCv2,
+Fan
+et
+al. [20] and Akhtar et al. [21] proposed to encode inter resid-
+uals between temporal successive PCG frames for dynamic
+PCGC. Their main difference lies in the generation of inter
+prediction signals. Fan et al. [20] used a SparseCNN-based
+motion estimation to align the coordinate of the reference to
+the current frame, and then interpolate k nearest neighbors
+to first derive the temporal prediction and then compute the
+residual difference; while Akhtar et al. [21] employed a “con-
+arXiv:2301.12165v1  [cs.CV]  28 Jan 2023
+
+2
+PCG t
+PCG t-1
+Nth 
+scale
+N-1th 
+scale
+SOPA
+(1-stage)
+m-1th 
+scale
+mth 
+scale
+Lossless Phase
+Lossy Phase
+Nth 
+scale
+N-1th 
+scale
+m-1th 
+scale
+mth 
+scale
+Extractor
+m-2th 
+scale
+m-2th 
+scale
+SparsePCGC
+Down-
+scaling
+Down-
+scaling
+Predictor
+SConv 93×32
+Extractor
+Extractor
+Extractor
+Predictor
+SConv 93×32
+Predictor
+SConv 93×32
+Predictor
+SConv 93×32
+Encoder
+Encoder
+SOPA
+(1-stage)
+SOPA
+(8-stage)
+SOPA
+(8-stage)
+feat
+feat
+feat
+feat
+feat
+(a)
+Encoder or
+Feature Extractor
+SOPA (1-stage)
+Scale i
+Scale i-1
+Scale i-1
+Scale i
+Occuancy 
+Probability
+(b)
+Fig. 1: Dynamic PCGC in a two-frame example. (a) On top of the MSR framework used by SparsePCGC for static PCGC
+originally, multiscale temporal priors of (t−1)-th frame are first extracted using Extractors and transferred using Predictors for
+the compression of t-th frame, where temporal priors are concatenated with the same-frame lower-scale priors for improving
+the capacity of SOPA model. (b) Network examples for Encoder (or Feature Extractor) and 1-stage SOPA. Lossy SparsePCGC
+is comprised of a lossless phase using 8-stage SOPA and a lossy phase using 1-stage SOPA instead, across different scales.
+On the contrary, lossless SparsePCGC uses 8-stage SOPA for all scales [1]. Sparse Convolution (SConv) constitutes the basic
+feature processing layer. Inception-ResNet (IRN) blocks are used for deep feature aggregation [2].
+volution on target coordinates” operation to map the feature-
+space information from the reference to the current frame to
+derive the inter residual.
+This letter also applies the “convolution on target coor-
+dinates” to exploit correlations across temporal successive
+frames in feature space. Instead of using the inter residual
+at a fixed scale, we generate multiscale temporal priors for
+scale-wise contextual information aggregation, which greatly
+improves the conditional probability approximation in com-
+pression of our method, as shown subsequently.
+III. PROPOSED METHOD
+A. Overall Framework
+The proposed MSR-based dynamic PCGC is shown in
+Fig. 1. A two-frame example is illustrated where the (t − 1)-
+th frame is already encoded and reconstructed as the temporal
+reference, and the t-th frame is about to be encoded. Appar-
+ently, such a two-frame example can be easily extended to a
+sequence of frames.
+To compress the t-th frame, a straightforward solution is
+to encode each PCG frame independently, a.k.a. intra coding,
+using default SparsePCGC to solely exploit cross-scale and
+same-scale correlations in the same frame. As there are strong
+temporal correlations across successive frames, inter prediction
+is often utilized for improving compression efficiency. To this
+end, this work follows the MSR principle to first progressively
+extract features using Extractors from the (t − 1)-th recon-
+struction ˆxt−1, and then generate multiscale temporal priors
+via a one-layer sparse convolution (SConv) based Predictors
+for inter conditional coding of t-th frame xt.
+Similar to the SparsePCGC, dyadic resampling is applied
+for multiscale computation [1]. Assuming the highest scale
+of an input point cloud at N, the lossy compression of this
+PCG is comprised of m-scale lossless and (N −m)-scale lossy
+compression. Adapting m is to balance the lossy rate-distortion
+tradeoff [22]. As seen, in the lossless phase, temporal priors
+from the inter reference are concatenated with the lower-
+scale spatial priors in the same intra frame which are then
+fed into the 8-stage SOPA model for better approximation of
+occupancy probability for lossless coding; while in the lossy
+phase, such concatenated spatiotemporal priors can be either
+augmented with decoded local neighborhood information or
+directly used in 1-stage SOPA model for better geometry
+reconstruction. By contrast, the lossless compression of an
+input PCG applies 8-stage SOPA uniformly for all scales to
+process such concatenated spatiotemporal priors.
+We next detail each individual module developed for the
+use of multiscale temporal priors in inter conditional coding.
+B. Encoder/Extractor & SOPA Models
+The Encoder (and Extractor) model which is typically
+devised with the resolution downscaling, aggregates local
+neighborhood information to form spatial intra (or temporal
+inter) priors for enhancing the SOPA model. Correspondingly,
+the SOPA model estimates the occupancy probability for ge-
+ometry reconstruction (i.e., voxel occupancy status) gradually
+from lower to higher scale, using both spatial priors (e.g.,
+decoded latent feature, lower-scale input) in the same frame
+and temporal priors from the inter reference.
+The Encoder/Extractor model applies sparse convolutions
+and nonlinear activations for computation as shown in Fig. 1b,
+consisting of a convolutional voxel downsampling layer with
+kernel size and stride of 2 at each dimension, e.g., SConv 23×
+32 s2↓, and stacked Inception-ResNet (IRN) blocks for deep
+feature aggregation. The IRN contains multiple convolutional
+layers with a kernel size of 3×3×3, e.g., SConv 33 × 32 [2].
+The 1-stage SOPA model mostly mirrors the processing of
+the Encoder/Extractor where a transposed convolutional voxel
+upsampling layer with kernel size and stride of 2 is used, e.g.,
+SConv 23 ×32 s2↑. This 1-stage SOPA can be easily extended
+to support multi-stage computation by grouping upscaled
+
+3
+������������������������
+������������������������
+�������������������������
+������������������������
+SOPA
+Encoder
+(a)
+������������������������
+������������������������
+������������������������−1
+������������������������
+�������������������������
+������������
+�������������������������−1
+������������
+Extractor
+Encoder
+SOPA
+(b)
+������������������������
+������������������������
+������������������������−1
+�������������������������−1
+������������������������, ������������������������−1
+�������������������������
+c
+c
+Extractor
+Encoder
+SOPA
+(c)
+Fig. 2: Intra and Inter Coding of PCGs. (a) intra coding used
+in SparsePCGC [1], (b) inter residual coding used in [20], [21],
+(3) the proposed inter conditional coding.
+voxels for stage-wise processing [1]. As exemplified in the
+lossless phase of Fig. 1, 8-stage SOPA is used to progressively
+reconstruct the voxels by utilizing previously-processed, same-
+scale neighbors for better probability estimation.
+C. Inter Conditional Coding
+We use the Predictor to transfer information from the inter
+reference for the compression of the current frame. As in
+Fig. 1, the Predictor is implemented using a one-layer sparse
+convolution to perform the “convolution on target coordi-
+nates”, which has the same number of parameters and opera-
+tions as the normal convolution, except the target coordinates
+of its output can be customized. For instance, a sparse tensor
+is formulated using a set of coordinates ⃗C = {(xi, yi, zi)}i
+and associated features ⃗F = {⃗fi}i. The sparse convolution is
+formulated as :
+⃗f out
+u
+=
+�
+k∈N3(u, ⃗Cin) Wk ⃗f in
+u+k
+for
+u ∈ ⃗Cout,
+(1)
+where ⃗Cin and ⃗Cout are input and output coordinates in the
+reference frame and current frame, respectively. N3(u, ⃗Cin) =
+{k|u + k ∈ ⃗Cin, k ∈ N3} defines a 3D convolutional kernel
+centered at u ∈ ⃗Cout with offset k in ⃗Cin. ⃗f in
+u+k and ⃗f out
+u
+are
+corresponding input and output feature vectors at coordinate
+u+k ∈ ⃗Cin and u ∈ ⃗Cout, respectively. Wi is kernel weights.
+In this work, the Predictor takes each coordinate of the current
+frame as the center, aggregates, and transfers the colocated
+features at each scale in a 9 × 9 × 9 local window of the
+reference, e.g., SConv 93 × 32.
+The use of temporal priors yt−1 from the reference for inter
+prediction is exemplified in Fig. 2. As for a comparison, intra
+coding is also pictured in Fig. 2a. The inter residual coding
+scheme is used in [20], [21] where the feature residual between
+the reference yt−1 and current frame is encoded as in Fig. 2b.
+The residual compensation is usually limited at the first layer
+of the lossy phase because it requires the correct geometry
+information for augmentation. Having residual compensation
+in other lossy scales is impractical because incorrect geometry
+would severely degrade the reconstruction quality [2].
+By contrast, a simple-yet-effective spatiotemporal feature
+concatenation is applied to perform the inter conditional
+coding in Fig. 2c which is flexible and applicable to all scales
+under the MSR framework. As seen, the reference reconstruc-
+tion ˆxt−1 is used to generate scale-wise temporal priors which
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+bpp
+64
+66
+68
+70
+72
+74
+76
+78
+D1 PSNR (dB)
+average_100
+Ours
+SparsePCGC
+Fan et al.
+Akhtar et al.
+PCGCv2
+V-PCC
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+bpp
+64
+66
+68
+70
+72
+74
+76
+78
+D1 PSNR (dB)
+average_32
+Ours
+SparsePCGC
+Fan et al.
+Akhtar et al.
+PCGCv2
+V-PCC
+Fig. 3: Efficiency Comparison. Rate-Distortion (R-D) curves
+of different methods. 100 (left) and 32 (right) frames are eval-
+uated across a wide range of bitrates following the CTC [9].
+are then concatenated with the (cross-scale) spatial priors from
+the same frame to help the compression in both lossless and
+lossy compression. In this way, we retain all the information
+of temporal reference and use it for the compression of yt,
+which allows the codec to adaptively extract useful information
+for occupancy probability estimation. In lossless mode, it
+generates bitstream with less bitrate consumption; while in
+lossy mode, it helps to better reconstruct the geometry with
+less distortion.
+D. Loss Functions
+To quantify the voxel occupancy probability, we use the
+Binary Cross-Entropy (BCE) loss to measure the bitrate re-
+quired to encode the occupancy status. At the same time,
+the BCE loss also represents the geometry distortion in lossy
+compression. For the compression of latent feature in the
+encoder, we use a simple factorized entropy model [23] to
+estimate its probability, and cross-entropy loss to calculate the
+bitrate RF . The total loss function is the combination of BCE
+loss and rate consumption RF , i.e., Loss = BCE+λ·R, where
+λ is the weight used to adjust the rate-distortion tradeoff.
+IV. EXPERIMENTAL RESULTS
+A. Testing and Training Conditions
+Training and Testing Datasets. We use the 8i Voxelized
+Full Bodies (8iVFB) dataset [24] for training and the Owlii
+dynamic human sequence dataset [25] for testing. The training
+dataset contains 5 sequences: longdress, loot, redandblack,
+soldier, queen, each of which has 300 frames at 10-bit
+geometry precision. The test dataset contains 4 sequences:
+basketball player, dancer, model, exercise. They are all quan-
+tized to 10-bit geometry precision. The splitting of training
+and testing samples follows the Exploration Experiment (EE)
+recommendations used in MPEG AI-PCC group [9].
+Training Strategies. In training, we partition each frame
+into 4 blocks with kdtree and progressively downscale them
+to 4 different scales for data augmentation. We train one
+model for lossless coding and five models for lossy coding.
+By adjusting m in lossy phase and the R-D weight λ in the
+loss function, we obtain five different lossy coding models,
+covering bitrates from 0.01 to 0.18 bpp (bits per point).
+Testing Conditions. The testing follows the common test
+condition (CTC) defined in the AI-PCC group for dynamic
+PCGC [9]. The first frame is encoded in intra mode, followed
+by all P frames that use the temporally-closest reconstruction
+
+4
+TABLE I: Compression performance comparison with other methods (tested on 100/32 frames following the MPEG CTC [9])
+sequences
+(100/32)
+lossless (bpp)
+lossy (BD-Rate Gain %)
+G-PCC
+SparsePCGC
+Ours
+SparsePCGC
+Fan [20]
+Akhtar [21]
+PCGCv2 [2]
+V-PCC
+player
+0.824/0.812
+0.445/0.441
+0.400/0.388
+-27.7/-24.2
+-28.3/-28.0
+-49.8/-49.1
+-53.0/-51.3
+-78.6/-78.9
+dancer
+0.854/0.849
+0.461/0.460
+0.425/0.425
+-11.9/-14.0
+-26.7/-28.4
+-49.1/-50.0
+-45.8/-47.4
+-77.6/-79.2
+model
+0.840/0.811
+0.460/0.451
+0.404/0.388
+-28.9/-26.6
+-31.4/-31.7
+-50.2/-53.4
+-55.2/-55.0
+-76.8/-78.8
+exercise
+0.829/0/819
+0.448/0.442
+0.388/0.379
+-31.0/-25.3
+-26.0/-23.2
+-49.7/-47.7
+-55.3/-51.5
+-77.6/-77.2
+average
+0.837/0.823
+0.454/0.449
+0.404/0.395
+-24.9/-22.5
+-28.1/-27.8
+-49.7/-50.0
+-52.3/-51.3
+-77.7/-78.5
+TABLE II: Average runtime comparison in lossless mode
+Time (s/frame)
+G-PCC
+SparsePCGC
+Ours
+Enc
+4.75
+1.82
+1.96
+Dec
+2.60
+1.66
+1.82
+as the reference. Results are averaged for cases using 32
+frames and 100 frames. The bitrate is evaluated by the average
+bits per input point (bpp) for each sequence. The geometric
+distortion is evaluated by D1-PSNR per frame to produce a
+sequence-level average (the first intra frame is also included).
+B. Performance Evaluation
+For lossy coding, the V-PCC [26] is selected for comparison
+because of its SOTA performance for dynamic lossy PCGC;
+Here we apply the default low-delay HEVC video encoding
+in V-PCC. While for lossless coding, the G-PCC (octree) is
+compared because of its superior efficiency. Moreover, we
+compare with other learning-based PCGC methods, including
+the PCGCv2 [2] and the SparsePCGC [1] which were orig-
+inally developed for static PCGC, and two recently-emerged
+dynamic PCGC methods proposed by Akhtar et al. [21] and
+Fan et al. [20]. For the PCGCv2 and SparsePCGC, every
+PCG frame is coded independently as intra mode without
+inter prediction. Regarding learning-based methods [20], [21],
+since they are both being studied in the MPEG AI-PCC group
+following the CTC for training and testing [9], we directly cite
+their results reported in the latest standard ad-hoc summary for
+a fair comparison [27], [28].
+Comparison to G-PCC/V-PCC. As shown in Table I and
+Fig. 3, in lossless mode, the proposed method reaches an
+average 45% gain over the G-PCC anchor, e.g., 0.404 bpp
+versus 0.837 bpp when testing 100 frames; while in lossy
+mode, our method provides ≈78% BD-Rate improvement
+against the anchor V-PCC.
+Comparison to learned static PCGC. We present our BD-
+Rate gains over state-of-the-art learning-based methods used
+for static PCGC [2], [1]. As also in Table I, compared with
+PCGCv2 [2] that only supports the lossy coding, the proposed
+method attains 52.3%/51.3% BD-Rate reduction. In lossless
+mode, we improve the SparsePCGC [1] by around 11% on
+average (0.404/0.395 bpp versus 0.454/0.449 bpp); while in
+lossy mode, the gain over SparsePCGC is even higher, > 22%
+on average. Note that the proposed method is extended on top
+of the SparsePCGC by introducing inter conditional coding.
+The resultant BD-Rate gain further confirms the superiority of
+the use of multiscale temporal priors in dynamic PCGC.
+Comparison to learned dynamic PCGC. Further, we
+compare the proposed method with learning-based dynamic
+PCGC methods [20], [21] in Table I. We only compare lossy
+mode performance because their solutions only support lossy
+compression. As shown, the proposed method significantly
+outperforms existing methods with approximately 28% and
+50% BD-Rate gains over Fan et al. [20] and Akhtar et al. [21]
+on average. Our superior performance mainly attributes to: 1)
+we adopt a multi-stage SOPA in lossless phase, which is more
+efficient than the use of lossless G-PCC in [21], [20]; 2) in
+the lossy phase, inter residual compensation at a fixed scale
+limits the performance of [21], [20]. Note that even using the
+same lossless G-PCC in our method as in [20], [21], the BD-
+Rate gains are also mostly retained due to the use of inter
+conditional coding.
+We also visualize corresponding R-D curves in Fig. 3. It
+shows that our method consistently performs better than other
+methods across a wider range of bitrates. It is also observed
+that Fan et al. [20] focus on high bitrates and cannot reach
+at bitrates below 0.06 bpp, while Akhtar et al. [21] is mostly
+applicable to low bit rates but performs poorly at high bitrates.
+This occurs mainly due to the fixed scale setting in their
+respective lossy phase, i.e., Fan et al. [20] downscales 2
+times and Akhtar et al. [21] downscales 3 times, for lossy
+compression. By contrast, our method provides flexible scale
+adjustment (i.e. high/medium/low bitrates with adaptive m
+e.g., m ∈ 1, 2, 3), and multiscale inter conditional coding
+through simple-yet-effective feature concatenation. These im-
+provements not only enable the support of both lossless and
+lossy compression but also yield SOTA performance.
+Complexity. We collect the runtime by respectively running
+the G-PCC, SparsePCGC, and the proposed method in lossless
+coding, as shown in Table II for complexity evaluation. The
+runtime is tested on an Intel Xeon Silver 4210 CPU and
+an Nvidia GeForce RTX 2080 GPU, which is just used as
+the intuitive reference to have a general understanding of
+the computational complexity. As seen, the proposed method
+presents faster encoding and decoding than G-PCC when
+using GPU acceleration. The runtime increase relative to the
+SparsePCGC-based intra coding is marginal.
+V. CONCLUSION
+This paper presents the compression of dynamic point cloud
+geometry, which incorporates the multiscale temporal priors
+into the multiscale sparse representation framework to enable
+inter conditional coding across temporal frames. Extensive
+experiments demonstrate that the proposed approach achieves
+SOTA performance in both lossy and lossless modes when
+compressing the dense object point cloud geometry.
+
+5
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+

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+page_content='1  Dynamic Point Cloud Geometry Compression  Using Multiscale Inter Conditional Coding  Jianqiang Wang, Dandan Ding, Hao Chen, and Zhan Ma  Abstract—This work extends the Multiscale Sparse Repre-  sentation (MSR) framework developed for static Point Cloud  Geometry Compression (PCGC) to support the dynamic PCGC  through the use of multiscale inter conditional coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' To this  end, the reconstruction of the preceding Point Cloud Geometry  (PCG) frame is progressively downscaled to generate multi-  scale temporal priors which are then scale-wise transferred  and integrated with lower-scale spatial priors from the same  frame to form the contextual information to improve occupancy  probability approximation when processing the current PCG  frame from one scale to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Following the Common Test  Conditions (CTC) defined in the standardization committee, the  proposed method presents State-Of-The-Art (SOTA) compression  performance, yielding 78% lossy BD-Rate gain to the latest  standard-compliant V-PCC and 45% lossless bitrate reduction  to the latest G-PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Even for recently-emerged learning-based  solutions, our method still shows significant performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Index Terms—Dynamic point cloud geometry, Multiscale tem-  poral prior, Inter conditional coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' INTRODUCTION  Dynamic point clouds are of great importance for applica-  tions like holographic communication, autonomous machinery,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', for which the efficient compression of dynamic Point  Cloud Geometry (PCG) plays a vital role in service provision-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' In addition to rules-based Point Cloud Geometry Com-  pression (PCGC) technologies standardized by the ISO/IEC  MPEG (Moving Picture Experts Group), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', Video-based  PCC (V-PCC) and Geometry-based PCC (G-PCC) [3], [4], [5],  learning-based PCGC methods have been extensively investi-  gated in the past few years, greatly improving the performance  with very encouraging prospects [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Among those learning-  based solutions, multiscale sparse representation (MSR) [2],  [1], [7], [8] has improved the performance unprecedentedly by  effectively exploiting cross-scale and same-scale correlations  in the same frame of a static PCG for compact representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  The compression of a static PCG frame independently can also  be referred to as the intra coding of the PCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  This work extends the MSR framework originally developed  for static PCGs to compress the dynamic PCGs [2], [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' In this  regard, we suggest the inclusion of multiscale temporal priors  for inter conditional coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 1, for a previously-  reconstructed PCG frame (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', PCG t − 1), we progressively  downsample it and extract scale-wise hierarchical features  which are then transferred as additional temporal priors to help  the compression of the same-scale PCG tensor of the current  frame (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', PCG t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' To this end, we basically concatenate  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Chen and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Ma are with Nanjing University, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Ding  is with Hangzhou Normal University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  the same-scale temporal priors from the inter reference and  lower-scale spatial priors from the same intra frame to form  the contextual information for better conditional occupancy  probability approximation in compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Such an inter con-  ditional coding scheme for dynamic PCGC is implemented on  top of the SparsePCGC [1] originally developed for the static  PCGC, to quantitatively evaluate its efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Experimental  results demonstrate the leading performance of our method  when compared with existing methods (either rules-based  or learning-based ones) in both lossy and lossless modes,  following the Common Test Conditions (CTC) used in the  MPEG standardization committee [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' RELATED WORK  In addition to existing G-PCC and V-PCC standards and  other rules-based PCC methods in [10], [11], [12], [13], [14],  [15], an excessive number of learning-based PCC solutions  have emerged in the past years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Therefore the ISO/IEC MPEG  3D graphics coding group initiated the Artificial Intelligence-  based Point Cloud Compression (AI-PCC) to investigate po-  tential technologies for better compression of point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Static PCGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Recently, major endeavors have been paid  to study the compression of a static PCG [6], a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Static  PCGC, yielding voxel-based [16], [17], point-based [18],  octree-based [19], and sparse tensor-based approaches [7],  [2], [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Among them, sparse tensor-based methods not only  attain the leading performance but also have low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  The first representative work is the PCGCv2 [2] where a  static PCG tensor is hierarchically downsampled and lossily  compressed using a Sparse Convolutional Neural Network  (SparseCNN) based autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Later, the SparsePCGC [1]  improves the PCGCv2 greatly under a unified MSR framework  to support both lossless and lossy compression of various point  clouds by extensively exploiting cross-scale and same-scale  correlations for better contextual modeling using SparseCNN-  based Occupancy Probability Approximation (SOPA) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  More details regarding the MSR and SOPA model can be  found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Dynamic  PCGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  On  top  of  the  PCGCv2,  Fan  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [20] and Akhtar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [21] proposed to encode inter resid-  uals between temporal successive PCG frames for dynamic  PCGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Their main difference lies in the generation of inter  prediction signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [20] used a SparseCNN-based  motion estimation to align the coordinate of the reference to  the current frame, and then interpolate k nearest neighbors  to first derive the temporal prediction and then compute the  residual difference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' while Akhtar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [21] employed a “con-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='12165v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='CV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='28 Jan 2023  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='PCG t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='PCG t-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Nth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='scale  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='N-1th  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='scale  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='SOPA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='(1-stage)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
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+page_content='scale  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Lossless Phase  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Lossy Phase  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Nth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='scale  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
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+page_content='Extractor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='m-2th  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
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+page_content='scale  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='SparsePCGC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Down-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='scaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Down-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='scaling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Predictor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='SConv 93×32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Extractor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Extractor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Extractor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Predictor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='SConv 93×32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Predictor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='SConv 93×32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
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+page_content='Encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='SOPA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
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+page_content='SOPA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
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+page_content='feat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='feat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='(a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Encoder or  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Feature Extractor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='SOPA (1-stage)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Scale i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Scale i-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Scale i-1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Scale i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Occuancy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Probability  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 1: Dynamic PCGC in a two-frame example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' (a) On top of the MSR framework used by SparsePCGC for static PCGC  originally, multiscale temporal priors of (t−1)-th frame are first extracted using Extractors and transferred using Predictors for  the compression of t-th frame, where temporal priors are concatenated with the same-frame lower-scale priors for improving  the capacity of SOPA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' (b) Network examples for Encoder (or Feature Extractor) and 1-stage SOPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Lossy SparsePCGC  is comprised of a lossless phase using 8-stage SOPA and a lossy phase using 1-stage SOPA instead, across different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  On the contrary, lossless SparsePCGC uses 8-stage SOPA for all scales [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Sparse Convolution (SConv) constitutes the basic  feature processing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Inception-ResNet (IRN) blocks are used for deep feature aggregation [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  volution on target coordinates” operation to map the feature-  space information from the reference to the current frame to  derive the inter residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  This letter also applies the “convolution on target coor-  dinates” to exploit correlations across temporal successive  frames in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Instead of using the inter residual  at a fixed scale, we generate multiscale temporal priors for  scale-wise contextual information aggregation, which greatly  improves the conditional probability approximation in com-  pression of our method, as shown subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' PROPOSED METHOD  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Overall Framework  The proposed MSR-based dynamic PCGC is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' A two-frame example is illustrated where the (t − 1)-  th frame is already encoded and reconstructed as the temporal  reference, and the t-th frame is about to be encoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Appar-  ently, such a two-frame example can be easily extended to a  sequence of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  To compress the t-th frame, a straightforward solution is  to encode each PCG frame independently, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' intra coding,  using default SparsePCGC to solely exploit cross-scale and  same-scale correlations in the same frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As there are strong  temporal correlations across successive frames, inter prediction  is often utilized for improving compression efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' To this  end, this work follows the MSR principle to first progressively  extract features using Extractors from the (t − 1)-th recon-  struction ˆxt−1, and then generate multiscale temporal priors  via a one-layer sparse convolution (SConv) based Predictors  for inter conditional coding of t-th frame xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Similar to the SparsePCGC, dyadic resampling is applied  for multiscale computation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Assuming the highest scale  of an input point cloud at N, the lossy compression of this  PCG is comprised of m-scale lossless and (N −m)-scale lossy  compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Adapting m is to balance the lossy rate-distortion  tradeoff [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As seen, in the lossless phase, temporal priors  from the inter reference are concatenated with the lower-  scale spatial priors in the same intra frame which are then  fed into the 8-stage SOPA model for better approximation of  occupancy probability for lossless coding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' while in the lossy  phase, such concatenated spatiotemporal priors can be either  augmented with decoded local neighborhood information or  directly used in 1-stage SOPA model for better geometry  reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' By contrast, the lossless compression of an  input PCG applies 8-stage SOPA uniformly for all scales to  process such concatenated spatiotemporal priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  We next detail each individual module developed for the  use of multiscale temporal priors in inter conditional coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Encoder/Extractor & SOPA Models  The Encoder (and Extractor) model which is typically  devised with the resolution downscaling, aggregates local  neighborhood information to form spatial intra (or temporal  inter) priors for enhancing the SOPA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Correspondingly,  the SOPA model estimates the occupancy probability for ge-  ometry reconstruction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', voxel occupancy status) gradually  from lower to higher scale, using both spatial priors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=',  decoded latent feature, lower-scale input) in the same frame  and temporal priors from the inter reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  The Encoder/Extractor model applies sparse convolutions  and nonlinear activations for computation as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 1b,  consisting of a convolutional voxel downsampling layer with  kernel size and stride of 2 at each dimension, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', SConv 23×  32 s2↓, and stacked Inception-ResNet (IRN) blocks for deep  feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The IRN contains multiple convolutional  layers with a kernel size of 3×3×3, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', SConv 33 × 32 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  The 1-stage SOPA model mostly mirrors the processing of  the Encoder/Extractor where a transposed convolutional voxel  upsampling layer with kernel size and stride of 2 is used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=',  SConv 23 ×32 s2↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' This 1-stage SOPA can be easily extended  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='to support multi-stage computation by grouping upscaled  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='�������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='SOPA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='(a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='�������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������������������−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='�������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='�������������������������−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Extractor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='Encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='SOPA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='(b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������������������−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='�������������������������−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' ������������������������−1  �������������������������  c  c  Extractor  Encoder  SOPA  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 2: Intra and Inter Coding of PCGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' (a) intra coding used  in SparsePCGC [1], (b) inter residual coding used in [20], [21],  (3) the proposed inter conditional coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  voxels for stage-wise processing [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As exemplified in the  lossless phase of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 1, 8-stage SOPA is used to progressively  reconstruct the voxels by utilizing previously-processed, same-  scale neighbors for better probability estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Inter Conditional Coding  We use the Predictor to transfer information from the inter  reference for the compression of the current frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 1, the Predictor is implemented using a one-layer sparse  convolution to perform the “convolution on target coordi-  nates”, which has the same number of parameters and opera-  tions as the normal convolution, except the target coordinates  of its output can be customized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' For instance, a sparse tensor  is formulated using a set of coordinates ⃗C = {(xi, yi, zi)}i  and associated features ⃗F = {⃗fi}i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The sparse convolution is  formulated as :  ⃗f out  u  =  �  k∈N3(u, ⃗Cin) Wk ⃗f in  u+k  for  u ∈ ⃗Cout,  (1)  where ⃗Cin and ⃗Cout are input and output coordinates in the  reference frame and current frame, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' N3(u, ⃗Cin) =  {k|u + k ∈ ⃗Cin, k ∈ N3} defines a 3D convolutional kernel  centered at u ∈ ⃗Cout with offset k in ⃗Cin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' ⃗f in  u+k and ⃗f out  u  are  corresponding input and output feature vectors at coordinate  u+k ∈ ⃗Cin and u ∈ ⃗Cout, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Wi is kernel weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  In this work, the Predictor takes each coordinate of the current  frame as the center, aggregates, and transfers the colocated  features at each scale in a 9 × 9 × 9 local window of the  reference, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', SConv 93 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  The use of temporal priors yt−1 from the reference for inter  prediction is exemplified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As for a comparison, intra  coding is also pictured in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The inter residual coding  scheme is used in [20], [21] where the feature residual between  the reference yt−1 and current frame is encoded as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  The residual compensation is usually limited at the first layer  of the lossy phase because it requires the correct geometry  information for augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Having residual compensation  in other lossy scales is impractical because incorrect geometry  would severely degrade the reconstruction quality [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  By contrast, a simple-yet-effective spatiotemporal feature  concatenation is applied to perform the inter conditional  coding in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 2c which is flexible and applicable to all scales  under the MSR framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As seen, the reference reconstruc-  tion ˆxt−1 is used to generate scale-wise temporal priors which  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='5  bpp  64  66  68  70  72  74  76  78  D1 PSNR (dB)  average_100  Ours  SparsePCGC  Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Akhtar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  PCGCv2  V-PCC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='5  bpp  64  66  68  70  72  74  76  78  D1 PSNR (dB)  average_32  Ours  SparsePCGC  Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Akhtar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  PCGCv2  V-PCC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 3: Efficiency Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Rate-Distortion (R-D) curves  of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 100 (left) and 32 (right) frames are eval-  uated across a wide range of bitrates following the CTC [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  are then concatenated with the (cross-scale) spatial priors from  the same frame to help the compression in both lossless and  lossy compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' In this way, we retain all the information  of temporal reference and use it for the compression of yt,  which allows the codec to adaptively extract useful information  for occupancy probability estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' In lossless mode, it  generates bitstream with less bitrate consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' while in  lossy mode, it helps to better reconstruct the geometry with  less distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Loss Functions  To quantify the voxel occupancy probability, we use the  Binary Cross-Entropy (BCE) loss to measure the bitrate re-  quired to encode the occupancy status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' At the same time,  the BCE loss also represents the geometry distortion in lossy  compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' For the compression of latent feature in the  encoder, we use a simple factorized entropy model [23] to  estimate its probability, and cross-entropy loss to calculate the  bitrate RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The total loss function is the combination of BCE  loss and rate consumption RF , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', Loss = BCE+λ·R, where  λ is the weight used to adjust the rate-distortion tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' EXPERIMENTAL RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Testing and Training Conditions  Training and Testing Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' We use the 8i Voxelized  Full Bodies (8iVFB) dataset [24] for training and the Owlii  dynamic human sequence dataset [25] for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The training  dataset contains 5 sequences: longdress, loot, redandblack,  soldier, queen, each of which has 300 frames at 10-bit  geometry precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The test dataset contains 4 sequences:  basketball player, dancer, model, exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' They are all quan-  tized to 10-bit geometry precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The splitting of training  and testing samples follows the Exploration Experiment (EE)  recommendations used in MPEG AI-PCC group [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Training Strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' In training, we partition each frame  into 4 blocks with kdtree and progressively downscale them  to 4 different scales for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' We train one  model for lossless coding and five models for lossy coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  By adjusting m in lossy phase and the R-D weight λ in the  loss function, we obtain five different lossy coding models,  covering bitrates from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='18 bpp (bits per point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Testing Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The testing follows the common test  condition (CTC) defined in the AI-PCC group for dynamic  PCGC [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The first frame is encoded in intra mode, followed  by all P frames that use the temporally-closest reconstruction  4  TABLE I: Compression performance comparison with other methods (tested on 100/32 frames following the MPEG CTC [9])  sequences  (100/32)  lossless (bpp)  lossy (BD-Rate Gain %)  G-PCC  SparsePCGC  Ours  SparsePCGC  Fan [20]  Akhtar [21]  PCGCv2 [2]  V-PCC  player  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
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+page_content='2  average  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='837/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='823  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='454/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='449  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='404/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='395  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='9/-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='5  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='1/-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='8  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='7/-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='0  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='3/-51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='3  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='7/-78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='5  TABLE II: Average runtime comparison in lossless mode  Time (s/frame)  G-PCC  SparsePCGC  Ours  Enc  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='82  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='96  Dec  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='66  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='82  as the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Results are averaged for cases using 32  frames and 100 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The bitrate is evaluated by the average  bits per input point (bpp) for each sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The geometric  distortion is evaluated by D1-PSNR per frame to produce a  sequence-level average (the first intra frame is also included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Performance Evaluation  For lossy coding, the V-PCC [26] is selected for comparison  because of its SOTA performance for dynamic lossy PCGC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Here we apply the default low-delay HEVC video encoding  in V-PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' While for lossless coding, the G-PCC (octree) is  compared because of its superior efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Moreover, we  compare with other learning-based PCGC methods, including  the PCGCv2 [2] and the SparsePCGC [1] which were orig-  inally developed for static PCGC, and two recently-emerged  dynamic PCGC methods proposed by Akhtar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [21] and  Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' For the PCGCv2 and SparsePCGC, every  PCG frame is coded independently as intra mode without  inter prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Regarding learning-based methods [20], [21],  since they are both being studied in the MPEG AI-PCC group  following the CTC for training and testing [9], we directly cite  their results reported in the latest standard ad-hoc summary for  a fair comparison [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Comparison to G-PCC/V-PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As shown in Table I and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 3, in lossless mode, the proposed method reaches an  average 45% gain over the G-PCC anchor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='404 bpp  versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='837 bpp when testing 100 frames;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' while in lossy  mode, our method provides ≈78% BD-Rate improvement  against the anchor V-PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Comparison to learned static PCGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' We present our BD-  Rate gains over state-of-the-art learning-based methods used  for static PCGC [2], [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As also in Table I, compared with  PCGCv2 [2] that only supports the lossy coding, the proposed  method attains 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='3%/51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='3% BD-Rate reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' In lossless  mode, we improve the SparsePCGC [1] by around 11% on  average (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='404/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='395 bpp versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='454/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='449 bpp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' while in  lossy mode, the gain over SparsePCGC is even higher, > 22%  on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Note that the proposed method is extended on top  of the SparsePCGC by introducing inter conditional coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  The resultant BD-Rate gain further confirms the superiority of  the use of multiscale temporal priors in dynamic PCGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Comparison to learned dynamic PCGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Further, we  compare the proposed method with learning-based dynamic  PCGC methods [20], [21] in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' We only compare lossy  mode performance because their solutions only support lossy  compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As shown, the proposed method significantly  outperforms existing methods with approximately 28% and  50% BD-Rate gains over Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [20] and Akhtar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [21]  on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Our superior performance mainly attributes to: 1)  we adopt a multi-stage SOPA in lossless phase, which is more  efficient than the use of lossless G-PCC in [21], [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 2) in  the lossy phase, inter residual compensation at a fixed scale  limits the performance of [21], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Note that even using the  same lossless G-PCC in our method as in [20], [21], the BD-  Rate gains are also mostly retained due to the use of inter  conditional coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  We also visualize corresponding R-D curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' It  shows that our method consistently performs better than other  methods across a wider range of bitrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' It is also observed  that Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [20] focus on high bitrates and cannot reach  at bitrates below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='06 bpp, while Akhtar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [21] is mostly  applicable to low bit rates but performs poorly at high bitrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  This occurs mainly due to the fixed scale setting in their  respective lossy phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [20] downscales 2  times and Akhtar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' [21] downscales 3 times, for lossy  compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' By contrast, our method provides flexible scale  adjustment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' high/medium/low bitrates with adaptive m  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', m ∈ 1, 2, 3), and multiscale inter conditional coding  through simple-yet-effective feature concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' These im-  provements not only enable the support of both lossless and  lossy compression but also yield SOTA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' We collect the runtime by respectively running  the G-PCC, SparsePCGC, and the proposed method in lossless  coding, as shown in Table II for complexity evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The  runtime is tested on an Intel Xeon Silver 4210 CPU and  an Nvidia GeForce RTX 2080 GPU, which is just used as  the intuitive reference to have a general understanding of  the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' As seen, the proposed method  presents faster encoding and decoding than G-PCC when  using GPU acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' The runtime increase relative to the  SparsePCGC-based intra coding is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' CONCLUSION  This paper presents the compression of dynamic point cloud  geometry, which incorporates the multiscale temporal priors  into the multiscale sparse representation framework to enable  inter conditional coding across temporal frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' Extensive  experiments demonstrate that the proposed approach achieves  SOTA performance in both lossy and lossless modes when  compressing the dense object point cloud geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  5  REFERENCES  [1] Jianqiang Wang, Dandan Ding, Zhu Li, Xiaoxing Feng, Chuntong Cao,  and Zhan Ma, “Sparse tensor-based multiscale representation for point  cloud geometry compression,” IEEE Transactions on Pattern Analysis  and Machine Intelligence, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 1–18, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  [2] Jianqiang Wang, Dandan Ding, Zhu Li, and Zhan Ma,  “Multiscale  point cloud geometry compression,” 2021 Data Compression Conference  (DCC), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 73–82, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  [3] D Graziosi, O Nakagami, S Kuma, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', “An overview of ongoing  point cloud compression standardization activities: video-based (V-PCC)  and geometry-based (G-PCC),”  APSIPA Transactions on Signal and  Information Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 9, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content='  [4] Sebastian Schwarz, Marius Preda, Vittorio Baroncini, Madhukar Buda-  gavi, Pablo Cesar, Philip A Chou, Robert A Cohen, Maja Krivoku´ca,  S´ebastien Lasserre, Zhu Li, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=', “Emerging mpeg standards for point  cloud compression,” IEEE Journal on Emerging and Selected Topics in  Circuits and Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
+page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'}
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+Adsorption of melting DNA
+Debjyoti Majumdar1, ∗
+1Alexandre Yersin Department of Solar Energy and Environmental Physics, Jacob Blaustein Institutes for Desert Research,
+Ben-Gurion University of the Negev, Sede Boqer Campus 84990, Israel
+(Dated: February 1, 2023)
+The melting of a homopolymer double-stranded (ds) DNA is studied numerically, in the presence
+of an attractive and impenetrable surface on a simple cubic lattice. The two strands of the DNA are
+modelled using two self-avoiding walks, capable of interacting at complementary sites, thereby mim-
+icking the base pairing. The impenetrable surface is modelled by restricting the DNA configurations
+at the z ≥ 0 plane, with attractive interactions for monomers at z = 0. Further, we consider two
+variants for z = 0 occupations by ds segments, where one or two surface interactions are counted.
+This consideration has significant consequences, to the extent of changing the stability of the bound
+phase in the adsorbed state. Interestingly, adsorption changes to first-order on coinciding with the
+melting transition.
+Introduction: The denaturation of the double-stranded
+DNA (dsDNA) from a bound (ds) to an unbound single-
+stranded (ss) phase is an important step towards fun-
+damental biological processes such as DNA replication,
+RNA transcription, packaging of DNA and repairing [1].
+In vitro, the melting transition is induced by changing the
+temperature or pH of the DNA solution. However, the
+physiological condition would allow neither extremes of
+temperature nor pH level inside the cell. Therefore, the
+cell has to rely on other ambient factors to locally modify
+the stability of the ds structure of the DNA. Among oth-
+ers, one of the crucial factors and a potential candidate
+that can alter the stability of the native DNA form is in-
+teraction of the DNA with a surface, e.g., with proteins
+or cell membranes. The strands being polymers can un-
+dergo an adsorption transition, where the two strands,
+either in the ds or ss phase, get adsorbed on a surface
+[2]. In vivo, the protein-induced DNA-membrane com-
+plex is used during the replication process, cell division,
+and for inducing local bends in the rigid duplex DNA
+[3, 4].
+Again, adsorption is instrumental in packaging
+DNA inside the virus heads [5, 6]. On the technological
+front, the adsorbing property of the DNA is often used to
+target drug delivery in gene therapy [7, 8], and for man-
+ufacturing biosensors with quick and accurate detection
+of DNA in bodily samples.
+In all these instances, the
+surface-DNA interaction can be tuned by changing the
+nature of the surface. This tunability calls for a detailed
+phase mapping arising from the interaction of the DNA
+with the adsorbing surface.
+The melting and the adsorption transition individu-
+ally, forms the subject of many theoretical and experi-
+mental studies in the past. Theoretically, lattice mod-
+els have been useful in extracting sensible results on par
+with the experiments. The melting transition was shown
+to be first-order when excluded volume interactions are
+fully included [9]. On the other hand, the polymer ad-
+sorption transition was shown to be continuous [2, 10].
+With this in mind, in this paper, we explore the inter-
+play between the melting and the adsorption transition
+of a model homopolymer DNA, using a lattice adaptation
+of the Poland-Scheraga model on a simple cubic lattice.
+Self-avoidance is duly implemented among the intra- and
+inter-strand segments. We found that the melting vs. ad-
+sorption phase diagram is drastically different for the two
+different schemes of interaction between the ds and the
+adsorbing surface. For specific values of the coupling po-
+tentials, the two transitions overlap, with the continuous
+adsorption transition becoming first-order.
+The model: We model the DNA strands (say A and
+B) as two self-avoiding walks (SAWs), represented by the
+vectors rA
+i and rB
+j (1 ≤ i, j ≤ N), and capable of forming
+a base pair (bp) among the complementary monomers
+(i = j) from the two strands while occupying the same
+lattice site (rA
+i = rB
+i ). One end of the DNA is grafted in
+the z = 0 plane. The other end is free to wander in the
+z ≥ 0 direction, with the z = 0 plane impenetrable and
+attractive. An energy −ϵbp is associated with each bound
+bp independent of the bp index (homopolymer) and is
+represented by the reduced variable g = ϵbp/kBT, where
+T is the temperature and kB is the Boltzmann constant.
+For each interaction with the z = 0 surface, there is an
+energetic gain of −ϵs, represented by the reduced variable
+q = ϵs/kBT. Further, we consider two variants: model I
+and model II. The difference in the two variants is in the
+strength of the ds interaction with the surface; in model
+arXiv:2301.13272v1  [cond-mat.soft]  30 Jan 2023
+
+2
+(a)
+(b)
+Adsorbing
+surface
+(c)
+ds
+ss
+bubble
+Y-fork
+FIG. 1.
+(Color online) Schematic diagram for the (a) lat-
+eral view of model I, and (b) planar view of model II. In (a)
+representing model I, only one strand is interacting with the
+surface effectively in the bound state. While both the strands
+are simultaneously in contact in model II, as in (b). (c) Two-
+dimensional depiction of our lattice model.
+I, we consider only one unit of interaction (ϵs), while
+in model II, we consider two units of interaction (2ϵs),
+each for one of the strands. Such consideration comes
+from the speculation that when interacting sidewise, like
+in Fig. 1(a), there would be an effective interaction of
+one strand. By contrast, when both the strands touch
+the plane simultaneously, each strand would contribute
+[Fig. 1(b)]. These two scenarios may arise depending on
+the hardness of the surface. While metallic surfaces (such
+as Gold) used during experiments are hard, biological
+surfaces tend to be much softer. A schematic diagram
+of our model is shown in Fig. 1(c).
+The Hamiltonian
+for a typical configuration according to model II can be
+written as,
+βH = −g
+N
+�
+i=1
+δrA
+i ,rB
+i − q
+N
+�
+i=1
+�
+α=A,B
+δ0,zα
+i ,
+(1)
+where, β = 1/(kBT) and δi,j is the Kronecker delta. The
+adsorbing surface can generally be of complex geometry
+with different degrees of roughness and curvature. How-
+ever, we choose a smooth and impenetrable flat surface
+for simplicity. For simulation, we use the pruned and en-
+riched Rosenbluth method (PERM) to sample the equi-
+librium configurations, averaging over 108 tours. We set
+the Boltzmann constant kB = 1 throughout our study.
+For melting, the average number of bound bps per unit
+length (nc) serves as the order parameter with nc = 1 and
+0 in the bound and unbound phase, respectively. The
+bound and the unbound phases are dominated by energy
+and entropy, respectively, depending upon whichever
+minimizes the free energy.
+For our model, in the ab-
+sence of any adsorbing surface (i.e., q = 0), the melting
+takes place at gc = 1.3413 with the crossover exponent
+φm = 0.94 [9, 11]. On the other hand, the 3d to 2d ad-
+sorption of a lattice polymer is a continuous transition
+with the critical point at qc = 0.2856 [10]. For adsorp-
+tion, the average number of surface contacts per unit
+length (ns) is the order parameter [12], and we denote
+its fluctuation by Cs. The corresponding critical expo-
+nent controlling the growth of surface contacts at the
+critical point is φa, and the order parameter follows a
+scaling, ns ∼ N φa−1 [13]. The exponent φa is expected
+to be universal, and the most recent improved estimate of
+the critical exponent from computer simulations suggest
+φa = 0.48(4) [10, 14].
+Naively, one would expect four distinct phases when
+melting and adsorption are considered together [4]. How-
+ever, the unbound-adsorbed phase was found missing in a
+theoretical study [15], which employs a model similar to
+model II, except that excluded volume interactions were
+neglected. Overall, in Ref. [15], it was found that the
+bound state is stabilized in the presence of an adsorbing
+surface. By contrast, on the experimental side, Ref. [16]
+had demonstrated that directly adsorbed DNA hybrids
+are significantly less stable than if free. Therefore, fur-
+ther study of the melting-adsorption interplay, employing
+more versatile models is essential for a complete under-
+standing.
+Model I: In this model variant, we consider equal sur-
+face interaction energy for both ss and ds segments.
+This choice of interaction yields four equilibrium phases,
+viz., bound-desorbed (BD), unbound-desorbed (UD),
+unbound-adsorbed (UA), and the bound-adsorbed (BA)
+phase [Fig. 2(a)]. The melting and the adsorption lines
+are obtained by varying g and q, respectively, while keep-
+ing one of them fixed [13]. The error bars in qc and gc
+are of the size of the plotting points. As the two lines
+(gc = 1.3413 and qc = 0.2856) approach each other, the
+bound state is primarily stabilized for increasing q, which
+is somewhat surprising [Fig. 2(c)]. This increased stabil-
+ity of the bound state persists for 0.26(6) <∼ q <∼ 0.4, and
+is perhaps due to the fact, that, in this region the bound
+and unbound phases in the vicinity of the melting line
+are unequally placed in the adsorbed phase. This short
+period of stability is followed by a steady increase in the
+threshold g for bound state for q > 0.4, separating the
+destabilized bound and unbound state in the adsorbed
+phase. One can understand this using the energy-entropy
+
+3
+ 0.9
+ 1
+ 1.1
+ 1.2
+ 1.3
+ 1.4
+ 1.5
+ 1.6
+ 1.7
+ 1.8
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+BD
+UD
+UA
+BA
+(a)
+(b)
+(c)
+(d)
+g
+q
+melt
+ads
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+PnsX(2N)
+ns/(2N)
+700
+800
+900
+1000
+ 1.28
+ 1.3
+ 1.32
+ 1.34
+ 1.36
+ 1.38
+ 1.4
+ 0.28  0.32  0.36  0.4
+-20 -15 -10 -5
+ 0
+ 5  10  15  20
+ 0
+ 0.02
+ 0.04
+Cs/N2Φa-1
+(q-qc)NΦa
+1000
+900
+800
+700
+FIG. 2. (Color online) (a) Model I phase diagram for melting
+‘melt’ and adsorption ‘ads’ . The different phases are: bound-
+desorbed (BD), unbound-desorbed (UD), unbound-adsorbed
+(UA), and bound-adsorbed (BA). The dotted lines represent
+the transition points for the individual cases; for melting gc =
+1.3413 and for adsorption qc = 0.2856. (b) Scaling plots of
+the probability distribution (Pns) of surface contacts (ns) on
+the BA → UA transition line corresponding to g = 1.5 and
+qc = 0.659, and for chain lengths N = 700 to 1000. (c) A
+zoom in of the phase diagram in (a) showing a decrease in
+the threshold g for bound state. (d) Scaling plot of surface
+contact fluctuation Cs for g = 1.5, using qc = 0.659 and
+φa = 0.99.
+argument; since the number of independent surface con-
+tacts increases upon unbinding, with each ds bp result-
+ing in two new possible ss surface contacts, along with
+an increase in the entropy, the UA phase is strongly fa-
+vored over the BA phase. A significant consequence is,
+the melting in the adsorbed phase (BA→UA) is differ-
+ent from the pure melting in two-dimensions (2d) where
+the melting point is at gc = 0.753(3). Noticeably, while
+undergoing UA to BA transition by varying q, the sys-
+tem shows first-order like fluctuation of surface contacts
+while the average number of surface contacts ns reduces
+to half its value than that in the UA phase [Fig. 2](d).
+This observation is supported by the scaling plot of the
+surface contact probability distribution (Pns) at a point
+(g = 1.5 and q = 0.659) above the melting phase bound-
+ary, using the scaling exponent φa = 0.99 for data col-
+lapse [Fig. 2](b). However, it is not a genuine desorption
+transition, and is due to the fact that the ds and ss sur-
+face contacts are treated on equal footing. For higher g
+values, the BA phase undergoes a continuous desorption
+around limg→∞ qc = 0.2856.
+Summarizing the results of model I, we see, that the
+bound phase is stabilized only for a small range of q val-
+ues [Fig. 2(c)].
+Otherwise, the bound state is mainly
+destabilized. For q < 0.265(5), the two transitions re-
+main decoupled without affecting each other.
+Results
+involving model I is in accordance with Ref. [16], where
+adsorbed DNA hybrids are found to be less stable than
+their free counterpart. Importantly, these results suggest
+that since the destabilization of the dsDNA is essential
+for the ease of opening up a bound segment, adsorption
+could play a crucial role in initiating certain biological
+processes related to the transferring of genetic informa-
+tion.
+Model II: For model II, a ds bound segment has a
+higher energy gain (precisely, double) than a ss segment
+upon interaction with the surface. Using this scheme of
+interaction, the phase plane is divided into four distinct
+phases viz., BD, UD, UA and the BA phase [Fig. 3].
+We can further identify three types of melting transition
+using these four phases: (i) when both the phases are
+desorbed, (ii) when the bound phase is adsorbed, and
+the unbound phase is desorbed, and (iii) when both the
+phases are adsorbed. While in the phases correspond-
+ing to the melting type (i) and (iii), the two transitions
+remain decoupled, for melting type (ii), both the tran-
+sitions coincide into one transition, represented by an
+overlapping phase boundary giving rise to multicritical
+points.
+Intriguingly, the adsorption transition is pro-
+moted to first-order in this overlapping region.
+Adja-
+cent to this overlapping region, and bounded by the lines
+g = 1.3413 and q = 0.2856 on the other two sides, is
+a small triangular island (denoted by a) [Fig. 3], akin
+to the Borromean phase found in nuclear systems [15].
+This a phase is not possible when either of the poten-
+tials is turned off, and exists as a result of the combined
+effect of the two potentials, even though neither g nor q
+is strong enough to support an ordered state, individu-
+ally. This small window of q and g values, corresponding
+to the coinciding phase line, facilitates achieving an ad-
+sorbed and a bound phase by changing only g or q, with
+the other parameter fixed. Such points (or region) can
+be crucial for real biological systems since it reduces a
+multi-parameter system to be controlled by a single pa-
+rameter. Adsorption in this region follows the same scal-
+
+4
+ 0.6
+ 0.8
+ 1
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+BD
+UD
+UA
+BA
+a
+(ar1)
+(a)
+(b)
+(c)
+g
+q
+melt
+ads
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 0
+ 0.1  0.2  0.3  0.4  0.5  0.6  0.7
+PnsX(2N)
+ns/(2N)
+500
+600
+700
+800
+900
+1000
+-6 -4 -2  0  2
+ 4  6
+ 0
+ 0.02
+ 0.04
+ 0.06
+ 0.08
+ 0.1
+Cs/N2φa-1
+(q-qc)Nφa
+1000
+900
+800
+700
+FIG. 3. (Color online) (a) Model II phase diagram. The dif-
+ferent phases are: bound-desorbed (BD), unbound-desorbed
+(UD), unbound-adsorbed (UA) and bound-adsorbed (BA).
+Dashed lines represent, g = 0.753 in red and q = 0.1428 in
+gray. Dotted lines represent, g = 1.3413 and q = 0.2856. (b)
+Probability distribution of surface contacts (Pns) at gc = 1.25
+and qc = 0.278 (arrow ar1 in (a)), for chain lengths N = 700
+to 1000. (c) Scaling plot for fluctuation of average number
+of surface contacts per unit length Cs for g = 1.25 using
+φa = 0.98 and qc = 0.277(7).
+ing exponent as of the first-order melting transition with
+φa = φm ∼ 1 [Fig. 3(c)] [17]. A first-order adsorption
+is also evident from the probability distribution of the
+surface contacts (Pns) at the transition point, e.g., for
+gc = 1.25 and qc = 0.278 in Fig. 3(b) [18]. The melting
+transition, however, remains unaffected. Below a, the ad-
+sorbed phase is destabilized for a small range of g values.
+For the transition from BA to UA phase the melting is
+two dimensional for sufficiently large q with φm ≈ 1.5
+when the system is completely adsorbed.
+Unlike model I, the bound state in model II is stabi-
+lized in the presence of the adsorbing surface. Since, post-
+melting, the entropy gain is smaller in the adsorbed phase
+(two dimensions), compared to the unbound state in the
+desorbed phase (three dimensions), the bound state in
+the adsorbed phase is more stable than that in the des-
+orbed phase, leading to a gradual lowering in the thresh-
+old g, which finally converges to limq→∞ gc ≈ 0.753(3),
+the two-dimensional melting point. A similar argument
+also applies for the adsorption transition for which the
+critical adsorption strength qc decreases and saturates at
+limg→∞ qc = 0.1428 [20].
+Although our results from model II are in line with
+Ref. [15], qualitatively, we obtain all four possible phases,
+instead of three, as in [15], where the UA phase was
+absent. Biologically, adsorption-induced stability could
+be important to guard DNA native form against thermal
+fluctuation and external forces. Importantly, adsorption
+can energetically compensate for the bending of the rigid
+ds segments, thereby, providing an alternative to bubble
+mediated bending [21].
+Conclusion: To conclude, in this paper, we elucidate
+the role of adsorption in modifying the melting transi-
+tion and vice-versa. Two separate models were consid-
+ered, which differs in the strength of interaction with
+the surface along the ds segments.
+Such a considera-
+tion arises from the speculation that the orientation of
+the DNA in conjunction with the nature of the adsorb-
+ing surface could play an important role in determining
+which of the studied model effectively applies. The two
+models show significant differences: model I shows that
+the ds structure is mostly destabilized in the presence
+of an attractive surface. This finding resemble the re-
+sult from the experiment performed with DNA hybrids
+in Ref. [16]. On the other hand, model II shows that DNA
+is only stabilized in the presence of an attractive surface.
+Although this model is similar to the theoretical model
+of Ref. [15], there are significant improvements, such as
+we consider excluded volume interaction. Moreover, we
+found the presence of all four possible phases, which is not
+the case in Ref [15]. In both the models, adsorption coin-
+ciding with the melting transition is first-order, however,
+whether this denotes a non-universality in the adsorp-
+tion transition is yet to be understood. Findings from
+both the models carry biological significance. Our work,
+therefore, contributes toward completing the picture by
+connecting the experimental and theoretical findings.
+Acknowledgement:
+D.M.
+was
+supported
+by
+the
+German-Israeli Foundation through grant number I-
+2485-303.14/2017 and by the Israel Science Foundation
+through grant number 1301/17, and the BCSC Fellow-
+ship from the Jacob Blaustein Center for Scientific Co-
+operation. Part of the simulations were carried out on the
+Samkhya computing facility at the Institute of Physics,
+Bhubaneswar.
+
+5
+∗ [email protected]
+[1] T. E. Cloutier and J. Widom, Mol. Cell 14, 355 (2004); J.
+Yan and J. F. Marko, Phys. Rev. Lett. 93, 108108 (2004).
+[2] E. Eisenriegler, K. Kremer and K. Binder, J. Chem. Phys.
+77, 6296 (1982).
+[3] W. Firshein, Annu. Rev. Microbiol., 43 89 (1989).
+[4] R. Kapri and S. M. Bhattacharjee, Eur. Phys. Letts. 83
+68002 (2008); R. Kapri, J. Chem. Phys. 130, 145105
+(2009).
+[5] G. A. Carri and M. Muthukumar, Phys. Rev. Lett. 82,
+5405-5408 (1999).
+[6] P. K. Purohit, et al., Biophys. Jour. 88, 851–866 (2005).
+[7] S. Z. Bathaie et al., Nucleic Acids Res. 27, 1001 (1999).
+[8] J. O. R¨adler et al., Science 275, 810 (1997).
+[9] M. S. Causo, B. Coluzzi, and P. Grassberger, Phys. Rev.
+E 62, 3958 (2000).
+[10] P. Grassberger, J. Phys. A: Math. Gen. 38, 323-331
+(2005).
+[11] φ = 1 for first-order transition, and φ < 1 for continu-
+ous/second order transition.
+[12] Here, length N denotes the maximum number of possible
+bps.
+[13] See Supplemental Material.
+[14] C. J. Bradly, A. L. Owczarek and T. Prellberg, Phys.
+Rev. E 97, 022503 (2018).
+[15] A. E. Allahverdyan et. al, Phys. Rev. Lett. 96, 098302
+(2006); A.E. Allahverdyan et. al, Phys. Rev. E 79, 031903
+(2009).
+[16] S. M. Schreiner et al., Anal. Chem. 83, 4288–4295 (2011).
+[17] A similar inter-change of the transition order was previ-
+ously observed in a theoretical model studying the inter-
+play of helix-coil transition and adsorption in a polymer
+[5].
+[18] A growing peak on either side of the distribution, and
+a deepening valley in between, is typical of a first-order
+transition. The valley represent suppressed states due to
+the growing surface term between the two phases. The
+inter-peak gap converges to a non-zero value. However, for
+models where this surface/interface, separating two coex-
+isting phases, is reduced to a point, this valley is absent
+[19]. Also see SM [13].
+[19] T. Garel, H. Orland, and E. Orlandini, Eur. Phys. J. B
+12, 261-268 (1999).
+[20] This is exact (other digits omitted) and can be obtained
+considering the fact, that, for model II even though the
+length is halved in the bound state, the energy in the
+adsorbed phase remains same. Therefore, the effective ad-
+sorbed energy per unit length (N) is doubled.
+[21] Double-stranded (ds) bound DNA segments are about 25
+times rigid than the single-stranded (ss) unbound DNA
+segments. These ss segments flanked by ds segments on
+either side are known as bubbles. These bubbles can act as
+hinge for bends in DNA.
+SUPPLEMENTARY MATERIAL
+I. SIMULATION ALGORITHM
+-60
+-40
+-20
+ 0
+ 20
+ 40-60
+-40
+-20
+ 0
+ 20
+ 40
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+S1
+S2
+X
+Y
+Z
+FIG. S1.
+(Color online) A typical configuration showing
+strand A (S1) and strand B (S2) with the adsorbing plane
+at z = 0.
+We use the pruned and enriched Rosenbluth algorithm
+(PERM) [1] to simulate the configurations of the dsDNA
+over an attractive surface [Fig. S1].
+Two strands are
+grown at once, adding monomers on the top of the lastly
+added monomer of both the strands at once. At each
+step, we calculate the joint possibilities of stepping into
+free sites obtained by a Cartesian product of the indi-
+vidual sets of possibilities i.e.
+Sn = Sn(A) × Sn(B).
+Each element in Sn corresponds to an ordered pair of
+new steps for both the strands, and carries a Boltzmann
+weight of exp(g × l + q × k), where l = 1 for a base-
+pair (bp) and 0 otherwise, while k = 0, 2 or 1 depending
+upon the number of surface contacts and model. Then,
+a choice is made according to the importance sampling.
+At each step the local partition function is calculated as
+wn = �
+Sn exp(g×l+q×k). The partition sum for length
+n is then estimated by product over the local partition
+sums at each step, Wn = �n
+i=1 wi, and averaging over
+the number of started tours, Zn = ⟨Wn⟩. Enrichment
+and pruning at nth step is performed depending on the
+
+6
+ratio, r = Zn/Wn:
+r =
+�
+�
+�
+�
+�
+1,
+continue to grow
+< 1,
+prune with probability (1 − r)
+> 1,
+make k-copies.
+If r < 1 and pruning fails, the configuration is contin-
+ued to grow but with Wn = Zn. For enrichment (r > 1)
+k is chosen as, k = min(⌊r⌋, N(Sn)), where each copy
+carries a weight Wn
+k , and N(Sn) is the cardinality of the
+set Sn. Averages are taken over 108 tours.
+At length n, any general thermodynamic observable
+(Qn) is averaged on the fly using the formula:
+⟨Qn⟩(g, q) = ⟨QnWn(g, q)⟩
+Zn(g, q)
+,
+(S1)
+where the ⟨· · · ⟩ in the numerator represents the run-
+ning average of the quantity over number of started tours
+and using the local estimate of the configuration weight
+Wn.
+One of the important aspects in simulating lattice
+self-avoiding walks is in checking if the immediate next
+sites are empty. The straightforward way is to check if
+any of the last N − 1 steps occupy the site. However,
+for walks of length N the time required in this oper-
+ation grows as O(N), and O(N 2) for the total chain.
+This can be avoided using the bit map method in which
+the whole lattice is stored in an array using a hashing
+scheme where each site is given an array address like:
+f(x, y, z) = x+yL+zL2 +offset, where L is the dimen-
+sions of the virtual lattice box and offset = ⌊Ld/2⌋ is a
+constant number which depends upon L to make the ad-
+dress start from zero. Here, the checking of self-avoidance
+is ≈ O(1), with no possibility of hashing collision. How-
+ever, since our problem requires constraining the polymer
+above the plane on which it is grafted there is a significant
+chance that the polymer will move out of the simulation
+box. A possible way out is to use a linked list method e.g.
+the AVL tree binary search [2]. In AVL, the algorithm
+works by creating a tree like structure where each node
+represent an occupied lattice site. Each entry for a new
+step is associated with search, insertion and rebalancing
+the tree branches. Each insertion or deletion operation
+requires O(log(n)) time, where n is the total number of
+nodes which translates to the number of monomers or oc-
+cupied sites or the polymer length. For a chain of length
+N + 1, the total growth time (assuming only insertion
+is performed) is: ln(1) + ln(2) · · · ln(N) = ln(N!). Using
+ 0
+ 0.05
+ 0.1
+ 0.15
+ 0.2
+ 0.25
+ 0.3
+ 0.35
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+Pn,nsX(2N)φa
+ns/(2N)φa
+500
+600
+700
+800
+900
+1000
+FIG. S2. (Color online) Scaling plot of surface contacts proba-
+bility distribution (Pn,ns) for different lengths N = 500−1000,
+at qc = 0.285 and g = 0.7 in model II. For data-collapse we
+use φa = 0.5. Notice that, while N is the number of max-
+imum possible bps, 2N is the maximum number of possible
+surface contacts.
+Sterling approximation, and for large N, this is approx-
+imately O(N ln N). Moreover, the AVL algorithm can
+be easily incorporated in the recursive structure of the
+PERM algorithm.
+II. SURFACE CONTACT HISTOGRAM
+Often, crossovers result into a melang´e of critical expo-
+nents, obtained from different methods such as the finite-
+size-scaling analysis, scaling of the specific heat peaks
+with length (N), the reunion exponent also known as the
+bubble-size-exponent (for DNA), among others.
+There-
+fore, deciding the behavior of the transition becomes dif-
+ficult. In this kind of situation it is advised to look at
+the probability distribution P(·) of the associated order
+parameter close to the transition point.
+A first-order transition is characterised by doubly
+peaked distribution with growing depth of the valley in
+between. This valley is the result of a d − 1 dimensional
+surface separating the two phases of the d dimensional
+system which suppresses the states in between the peaks.
+It grows exponentially deep in the thermodynamic limit,
+
+7
+P ∼ exp(−σLd−1), where L is the size of the system.
+However, for certain models (or problems) this interface
+can be reduced to a point separating the two phases e.g.
+in our DNA model the interface between a bound seg-
+ment and an unbound segment is a point, in adsorption
+a point separates the adsorbed and desorbed phases, or
+the point interface separating the collapse-ferromagnetic
+phase from the coiled-paramagnetic phase in the case of
+a magnetic polymer [3]. In these situations the valley is
+absent and the surface free energy is no longer extensive
+in N.
+To understand the change in the nature of the adsorp-
+tion transition, we look at the probability distribution
+of the surface contacts (ns) at different lengths, denoted
+by Pn,ns close to the transition point (qc). To calculate
+Pn,ns(q, g), we find the conditional partition sum Zn,ns
+for fixed q and g, where n is the length having ns number
+of surface contacts for different lengths. Finally, Pn,ns is
+found using the formula,
+Pn,ns(q, g) =
+Zn,ns(q, g)
+�2n
+ns=0 Zn,ns(q, g)
+.
+(S2)
+For a continuous transition, the order parameter distri-
+bution is expected to hold a scaling relation of the form
+Pns ∼ N −φap(ns/N φa).
+(S3)
+In Fig. S2, we show the scaling plot for Pn,ns for the
+adsorption transition in the unbound state corresponding
+to q = 0.285 and g = 0.7.
+III. ESTIMATION OF THE TRANSITION
+POINTS
+For q < qc, the partition sum of a SAW scales as
+Z(q, N) ∼ µNN γ1−1,
+(S4)
+where the subscript 1 in the entropic exponent γ1 de-
+notes the fact that one end is grafted on an impenetra-
+ble surface, while the exponential growth through µ (the
+effective coordination number) is invariant. Near the ad-
+sorption transition (q ∼ qc), Z(q, N) should scale as
+Z(q, N) ∼ µNN γ′
+1−1ψ[(q − qc)N φa],
+(S5)
+where ψ(x) is the scaling function.
+Taking derivative
+of ln Z(q, N) in Eq. (S5) with respect to q, and setting
+q = qc, one obtains the scaling form of the mean adsorbed
+energy per unit length (N) at the critical point as
+ns ∼ N φa−1.
+(S6)
+Therefore, at the critical adsorption point the quan-
+tity ns/N φa−1 should be N independent for N → ∞.
+For example, in Fig. S3(b) the estimated critical adsorp-
+tion point using Eq. (S6) is qc = 0.1431(5) for g = 5.
+For higher g’s, when the chain is completely bound, this
+should converge to qc = 0.1428.
+One must be careful
+to use the appropriate φa; for continuous transitions we
+use φa = 1/2, and φa = 0.92 for first-order transitions.
+We can have an idea about the nature of the transition
+and that about the transition point, beforehand, from
+the shape of the Cs curves. Further, following Ref. [4],
+we also looked at the quantity,
+γ′
+1,eff = 1 + ln
+�
+Z(q, 2N)/Z(q, N/2)/µ3N/2�
+ln 4
+,
+(S7)
+using µ = 4.6840386.
+Here, we simulate chains of
+length upto N = 10, 000, to see ns/N φa−1 and γ′
+1,eff
+upto N = 5000 [Fig. S4]. However, since our model has
+added complexities, e.g., two complementary monomers
+from different strands can occupy the same site to form a
+bp, we think that Eq. (S6) to be more reliable to estimate
+qc.
+For melting, we looked at the average number of bound
+bps per unit length (nc) and its fluctuation (Cc), to es-
+timate the transition points. The melting points are ob-
+tained from the scaling (or data collapse) of nc and Cc,
+following the equations,
+nc ∼ N φm−1f[(g − gc)N φm],
+(S8)
+and,
+Cc ∼ N 2φm−1h[(g − gc)N φm],
+(S9)
+Tuning gc and φm to the appropriate values would
+make the data for different lengths fall upon each other
+resulting in data collapse.
+For continuous adsorption transitions, we also use the
+crossing point of the Cs curves of the two longest lengths
+to determine the critical point [Fig. S3(a)]. However, for
+first-order adsorption the method of data collapse is used
+using Eq. (S8) and (S9) but with q in place of g and, nc
+and Cc replaced with ns and Cs, respectively.
+
+8
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 0.1
+ 0.12
+ 0.14
+ 0.16
+ 0.18
+ 0.2
+ 0.22
+Cs
+q
+1000
+800
+600
+400
+200
+ 3.4
+ 3.6
+ 100
+ 1000
+ns/N-0.5
+N
+q=0.1428
+q=0.1430
+q=0.1431
+q=0.1432
+q=0.1433
+FIG. S3. (Color online) (a) Fluctuation of surface contacts
+per unit length Cs for model II, g = 5, and lengths N =
+100 to 1000. (b) Long-length behavior of the average surface
+contacts per unit length (ns) scaled by N −0.5 for different
+q values around the critical adsorption point for g = 5 in
+model II. The adsorption transition is estimated to be qc =
+0.143 denoted by the dashed blue line in (a), and to be qc =
+0.1431(5) from (b).
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+ 100
+ 1000
+ns/NΦ-1
+N
+0.2700
+0.2800
+0.2856
+0.2870
+0.2900
+0.3000
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.01
+ 0.1
+γ'1,eff
+N-Φ
+0.2700
+0.2800
+0.2856
+0.2870
+0.2900
+0.3000
+FIG. S4. (Color online) Scaled average surface contacts per
+unit length ns/N −0.5 in (a) and γ′
+1,eff from Eq. (S7) in (b),
+using φ = 0.5 for different q values and g = 1.17 in model II.
+
+9
+∗ [email protected]
+[1] P. Grassberger,
+Pruned-enriched Rosenbluth method:
+simulations of θ polymers of chain length up to 1, 000, 000,
+Phys. Rev. E 56, 3682 (1997).
+[2] G. M. Adelson-Velsky and E. M. Landis, Dokl. Akad. Nauk
+SSSR 146, 263 (1962) [Soviet Math. Dokl, 3, 1259 (1962)].
+[3] T. Garel, H. Orland, and E. Orlandini, Eur. Phys. J. B
+12, 261-268 (1999).
+[4] P. Grassberger, J. Phys. A: Math. Gen. 38, 323-331
+(2005).
+

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+arXiv:2301.03112v1  [math.AT]  8 Jan 2023
+PERIODIC CYCLIC HOMOLOGY OVER Q
+KONRAD BALS
+Abstract. Let X be a derived scheme over an animated commutative ring of characteristic 0. We
+give a complete description of the periodic cyclic homology of X in terms of the Hodge completed
+derived de Rham complex of X. In particular this extends earlier computations of Loday-Quillen
+to non-smooth algebras. Moreover, we get an explicit condition on the Hodge completed derived
+de Rham complex, that makes the HKR-filtration on periodic cyclic homology constructed by
+Antieau and Bhatt-Lurie exhaustive.
+1. Introduction
+For a commutative ring k and a k-algebra R, the Hochschild homology HH(R/k) gives an element
+in the derived category D(k) of k. It has proven itself to be an interesting invariant, appearing for
+example in trace methods computing algebraic K-theory or in Connes’ non-commutative geometry.
+It was also Connes in [Con85] who constructed the cyclic structure on Hochschild homology to
+define negative cyclic homology HC−(R/k) := HH(R/k)hS1 and later periodic cyclic homology
+HP(R/k) := HH(R/k)tS1 and proving a relation between HC− of smooth functions on a manifold
+and de Rham cohomology of the manifold.
+Transferring Connes’ geometric interpretation into
+algebraic observations in [LQ84] Loday and Quillen compute the homotopy groups HC−
+∗ (R/k) in
+terms of algebraic de Rham cohomology in many cases. For the purpose of this paper passing here
+to the Tate-construction, they prove:
+Theorem 1.1 ([LQ84]). Assume Q ⊂ k commutative and R a smooth commutative k-algebra, then
+HP∗(R/k) ∼=
+�
+n∈Z
+H∗−2n
+dR
+(R; k)
+In this paper we give a generalization of this computation to the non-smooth and non-affine
+situation. By the classical observation that HH(R[S−1]/k) ≃ HH(R/k) ⊗R R[S−1] for every affine
+open Spec(R[S−1]) ⊂ SpecR Hochschild homology extends to a sheaf HHk in the Zariski1 topology
+on schemes over k (c.f. [WG91]). In fact, similarly we get a sheaf HPk extending periodic cyclic
+homology. We recall the details in Appendix A and write HH(X/k) := Γ(X, HHk) and HP(X/k) :=
+Γ(X, HPk).
+Moreover, Hochschild homology as a functor CAlg♥
+k → D(k) from discrete k-algebras to D(k) is
+left Kan extended from discrete polynomial algebras2 and, thus, further extends to a sifted colimit
+preserving functor from the category of animated commutative (i.e. simplicial commutative) k-
+algebras CAlgan
+k/. So putting both generalizations together and writing LΩ∗
+X/k for the derived de
+Rham complex of a derived scheme X over an animated Q-algebra k, we can state our main theorem
+in a great generality. In particular, if k is discrete and X = Spec(R) for a discrete k-algebra R, this
+gives new results on the periodic cyclic homology of ordinary algebras.
+Theorem 1.2. Given an animated commutative ring k with Q ⊂ π0(k) and X a derived k-scheme,
+we have
+HP(X/k) ≃
+�
+n∈Z
+�
+LΩ∗
+X/k[−2n]
+1In fact by [BMS19] 3.4. even in the fpqc topology via a different argument.
+2If P• is a simplicial resolution of the k-algebra R, it suffices to check that | HH(P•/k)| ≃ HH(R/k).
+1
+
+2
+KONRAD BALS
+where �
+LΩ∗
+X/k is the completion of LΩ∗
+X/k with respect to the Hodge filtration LΩ≥•
+−/k.
+The key ingredient in the proof is to understand how the Tate-construction behaves under the
+passage from smooth algebras to general or even animated algebras and it is this behavior that lets
+the product appear on the right hand side.
+In [Ant19] Antieau constructs the HKR-filtration on HP(X/k) with n-th associated graded
+�
+LΩ∗
+X/k[2n]. If k is an (animated) Q-algebra we can give a complete identification of this HKR-
+filtration in terms of the equivalence of Theorem 1.2 and we prove
+Theorem 1.3. In the situation of Theorem 1.2 the HKR-filtration on HP(X/k) corresponds to the
+ascending partial product filtration on �
+n∈Z �
+LΩ∗
+X/k[−2n], that is
+Fili
+HKR HP(X/k) ≃
+�
+n≤−i
+�
+LΩ∗
+X/k[−2n].
+In particular, the HKR-filtration is exhaustive, if and only if �
+LΩ∗
+X/k is (homologically) bounded
+above.
+This criterion will give us a large class of examples with exhaustive HKR-filtration. If k is a
+discrete Noetherian commutative Q-algebra and X an ordinary scheme of finite type over Spec k,
+then Bhatt gives in [Bha12] a concrete way to compute �
+LΩ∗
+X/k, which in particular lives in non-
+positive degrees (cf. Corollary 4.27. loc.cit.). Passing to filtered colimits we get
+Corollary 1.4. If k is a discrete Q-algebra and X an ordinary qcqs scheme over k, then the HKR-
+filtration on HP(X/k) is exhaustive.
+Furthermore, the analysis of the Tate-filtration in characteristic 0, which is reviewed in the
+Appendix B, also gives a description of the multiplicativity of the equivalence in the Theorem
+1.2.
+In general for an algebra A ∈ CAlgk there is just no algebra structure on �
+n∈Z A[−2n],
+however, for a (animated) commutative k-algebra R, the object HP(R/k) carries a natural structure
+of a commutative algebra in Modk.
+In section 4 we construct the corresponding multiplicative
+structure on �
+n∈Z �
+LΩ∗
+X/k[−2n]. On homotopy groups the induced graded ring structure comes from
+LΩ≤n
+X/k((t)). Note that there is the terminal topology on π∗ �
+LΩ∗
+X/k making the maps π∗ �
+LΩ∗
+X/k →
+π∗LΩ≤n
+X/k continuous. It is not Hausdorff because not every element is detected in some π∗LΩ≤n
+R/k.
+With this we can almost completely describe the graded ring π∗ HP(X/k) in terms of π∗ �
+LΩ∗
+X/k:
+Theorem 1.5. In the situation of Theorem 1.2, we can describe the homotopy groups HP∗(X/k)
+algebraically as
+HP∗(X/k) ∼=
+��
+n∈Z
+antn : an ∈ π∗+2n �
+LΩ∗
+X/k
+�
+with addition and multiplication given as
+��
+n∈Z
+antn
+�
++
+��
+n∈Z
+bntn
+�
+=
+�
+n∈Z
+(an + bn)tn
+��
+n∈Z
+antn
+�
+·
+��
+n∈Z
+bntn
+�
+=
+�
+n∈Z
+cntn
+where cn is a limit of the finite partial sums of �
+i+j=n ai · bj in the topology on π∗ �
+LΩ∗
+R/k
+3
+3This is sometimes called a net and explicitly means for every open U ∋ 0, there is a finite subset I0 ⊂ {i+j = n},
+such that for all finite subset J ⊂ {i + j = n} containing I0 we have cn − �
+(i,j)∈J ai · bj ∈ U.
+
+PERIODIC CYCLIC HOMOLOGY OVER Q
+3
+However, we want to immediately issue the warning that because the topology on π∗ �
+LΩ∗
+R/k is not
+Hausdorff, the element cn ∈ π∗ �
+LΩ∗
+R/k is not uniquely determined as a limit. To fully understand
+the homotopy groups HP∗(X/k) algebraically, one, furthermore, has to analyze the lim1-terms
+contributing to π∗ �
+LΩ∗
+R/k.
+1.1. Outline. We begin in section 2 with a formality statement for S1-actions in the derived cat-
+egory over rational algebras (Corollary 2.3) in order to recall a coherent version of the HKR-theorem
+in Proposition 2.7. This allows us to coherently compute HP for smooth algebras.
+In section 3 we will use the language of filtrations in order to generalize the computations for
+smooth algebras to arbitrary derived schemes and prove Theorem 1.2 (cf. Theorem 3.4). In particu-
+lar we will use the multiplicativity of the Tate-filtration. The Tate-filtration itself and its multiplic-
+ative structure in the rational setting will be reviewed in the Appendix B. Furthermore in section 3
+we will exploit the consequences for the HKR-filtration and prove Theorem 1.3 and Corollary 1.4.
+Finally, the last section (4) is completely devoted to the proof of Theorem 1.5.
+1.2. Notation. Throughout this note we are freely using the ∞-categorical language as developed
+in [Lur09] and [Lur16]. In particular, for a commutative ring k we identify the derived category D(k)
+with the category Modk := ModHkSp of Hk-module spectra and thus view it as a stably symmetric
+monoidal ∞-category. It comes with a canonical lax symmetric monoidal functor ι: Ch∗(k) → D(k)
+from the 1-category of chain complexes and we will constantly abuse notation by identifying C∗
+with ιC∗ for C∗ ∈ Ch∗(k).
+Moreover, we will use the 1-category CDGAk of commutative differential graded algebras over
+k. An object (C∗, d) ∈ CDGAk consists of a commutative graded k-algebra �
+i∈Z Ci of discrete
+R-modules with differentials d: Ci−1 → Ci for all i > 0 satisfying the Leibniz rule. There will be
+two orthogonal ways to view a CDGAk as an object in CAlgk, either with 0 differential or with
+differential d and we already warn the reader to not confuse those functors.
+In particular, for a commutative ring k and a commutative k-algebra R, we will generally view
+the de Rham complex Ω∗
+R/k as an object in CAlgk := CAlg(Modk), and if we want to view it as a
+CDGA over k we write ΩH
+R/k.
+Later in the paper, we need to talk about filtrations in a stable category C, by which we always
+mean decreasingly indexed, Z-graded filtrations, i.e. functors from Zop
+≤ into C. For a symmetric
+monoidal category C we equip the category Fil(C) := Fun(Zop
+≤ , C) with the Day convolution tensor
+product ⊗Day. The n-th associated graded grnF of F is given by the cofibre of the map F n+1 → F n.
+A splitting of a filtration F • ∈ Fil(C) consists of a collection (An)n∈Z together with an map of
+filtrations �
+n≥• An → F • inducing an equivalence on associated graded. In particular, a splitting
+(An) of F canonical gives an identification grnF ≃ An.
+Finally, to fix vocabulary, a filtration
+F • ∈ Fil(C) on F ∈ C is complete if lim F • ≃ 0 and is exhaustive if colim F • ≃ F. We write
+Fil∧(C) ⊂ Fil(C) for the full subcategory on complete filtrations and denote by (−)∧ its left adjoint.
+1.3. Acknowledgment. I would like to thank Achim Krause, Jonas McCandless and Thomas
+Nikolaus for helpful discussions on this topic. Finally, again I want to thank Thomas Nikolaus for
+bringing this project up. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
+Foundation) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster:
+Dynamics–Geometry–Structure and the CRC 1442 Geometry: Deformations and Rigidity.
+2. Formality over Q
+The explicit computations heavily rely on strong formality properties that hold if working over
+Q-algebras.
+In this section we will prove a strong version of the HKR-theorem for Hochschild
+homology. This enables us to establish a coherent versions of the Theorem 1.1 copied from [LQ84].
+
+4
+KONRAD BALS
+Throughout the first section, let k be a discrete commutative Q-algebra. The key ingredient is a
+formality statement of C∗(S1, k), due to [TV11].
+Construction 2.1. The multiplication S1 × S1 → S1 and the diagonal S1 → S1 × S1 exhibit S1
+as an associative bialgebra in spaces. Because the symmetric monoidal structure on S is cartesian,
+by the dual of [Lur15][Proposition 2.4.3.8] the coalgebra structure given by the diagonal refines
+to a cocommutative coalgebra structure.
+Now the functor C∗(−, k): S → D(k) from spaces to
+the derived category of k taking singular chains with coefficients in k refines via the Eilenberg-
+Zilber maps to a symmetric monoidal functor.
+Therefore C∗(S1, k) acquires the structure of a
+cocommutative bialgebra in D(k).
+Moreover, the functor ι: Ch∗(k) → D(k) from the 1-category of chain complexes to the ∞-
+category D(k) is lax symmetric monoidal and precisely restricts to a symmetric monoidal functor
+on the full 1-subcategory ChK−flat
+∗
+(k) of K-flat chain complexes. Thus the chain complex for ǫ in
+degree 1
+Λk(ǫ) := (k · ǫ
+0−→ k · 1)
+with multiplication ǫ2 = 0 and primitive comultiplication ∆ǫ = (ǫ⊗1+1⊗ǫ) gives a cocommutative
+bialgebra object in D(k) under the identification of Λk(ǫ) as an element in D(k).
+Proposition 2.2 ([TV11]). In this setting where k is a discrete Q-algebra, there is a natural equi-
+valence C∗(S1, k) ≃ Λk(ǫ) as cocommutative bialgebras in D(k) for ǫ primitive in degree 1.
+For completeness reasons we would like to include a proof here:
+Proof. Both objects C∗(S1, k) and Λk(ǫ) have canonical augmentations coming from S1 → ∗ in S
+and ǫ �→ 0 in Ch∗(k). We will in fact show, that they even agree as augmented cocommutative
+algebras in D(k). Using the adjunction (e.g. cf. [Lur16] Theorem 5.2.2.174)
+bar: Algaug(coCAlgD(k))
+coCAlgaug(D(k)) :cobar
+it satisfies to construct a map of (co-)augmented cocommutative coalgebras under the bar-functor.
+In fact the computation in [Ada56] show that for C∗(S1, k) the unit of the adjunction C∗(S1, k) →
+cobar(barC∗(S1, k)) is an equivalence, so that an identification of barC∗(S1, k) ≃ C∗(BS1, k) trans-
+lates to an identification of C∗(S1, k) under cobar. Therefore, we want to understand the cocommut-
+ative coalgebra structure of barC∗(BS1, k) or equivalently the dual commutative algebra structure
+on C∗(BS1, k), as both objects are of finite type. A choice of a generator in H2(BS1, k) gives
+a map k[x] := Free(k[−2]) → C∗(BS1, k) from the free commutative k-algebra on a generator x
+in degree −2. Because Q ⊂ k, on homotopy groups both sides are free on a generator in degree
+−2 and we have C∗(BS1, k) ≃ k[x] is free as a commutative algebra. Finally, translating back to
+cocommutative bialgebras, we can compute
+cobar(k[x])∨ ≃ (bark[x])∨ ≃
+�
+k ⊗k[x] k
+�∨
+by resolving k with the DGA (Λk[x](ǫ∨), dǫ∨ = x) for a primitive element ǫ∨ in degree −1. Thus
+�
+k ⊗k[x] k
+�∨ ≃ Λk(ǫ∨)∨ ≃ Λk(ǫ) for ǫ a dual basis to ǫ∨.
+□
+From now on, to shorten notation we set A := Λk(ǫ) for |ǫ| = 1 primitive.
+Corollary 2.3. For a rational discrete algebra k the categories Fun(BS1, D(k)) and ModAD(k) are
+equivalent as symmetric monoidal categories, where the symmetric monoidal structure on the latter
+comes from the coalgebra structure on A.
+4There is a gap in the proof of the cited reference as pointed out by [DH22], which could be fixed in the latest
+version (v4) of [BCN21].
+
+PERIODIC CYCLIC HOMOLOGY OVER Q
+5
+Proof. There is a symmetric monoidal equivalence Fun(BS1, D(k)) ≃ ModC∗(S1,k)D(k) as sym-
+metric monoidal categories, where the symmetric monoidal structure on the right hand side comes
+from the cocommutative bialgebra structure on C∗(S1, k). Thus the equivalence C∗(S1, k) ≃ A as
+cocommutative bialgebras gives a symmetric monoidal equivalence of their module categories (c.f.
+Proposition 2.2.1. in [Rak20]).
+□
+Remark 2.4. The above equivalence induces the identity on underlying objects.
+Thus, given a
+complex X ∈ D(k) equipping X with an action of S1 is equivalent to providing a module structure
+over A. Informally, this amounts to a map d: k · ǫ[1] ⊗ X ≃ X[1] → X and coherent homotopies
+witnessing d2 ≃ 0.
+Construction 2.5. Let CDGAk denote the 1-category of commutative differential graded algebras
+over k as introduced in the Notations. Forgetting the differential, there is a functor CDGAk →
+CAlgCh∗(k) sending (C∗, d) ∈ CDGA to �
+i∈Z Ci[i] ∈ CAlgCh∗(k) with 0 differential.
+In 1-
+categories now an action of A precisely corresponds to an ascending differential, such that this
+functor refines through CAlgModACh∗(k) and postcomposing with ι we get a map
+CDGAk → CAlgModACh∗(k) → CAlgModAD(k).
+To avoid confusion we will write U : CDGAk → CAlgBS1
+k
+for this functor.
+Remark 2.6. For a k-algebra R the de Rham complex ΩH
+R/k by definition lives in CDGAk. Now via
+the previous construction the underlying chain complex
+UΩH
+R/k ≃
+�
+n∈N
+Ωn
+R/k[n] ≃
+�
+· · ·
+0−→ Ω2
+R/k
+0−→ Ω1
+R/k
+0−→ Ω0
+R/k
+�
+(1)
+gives an object in CAlgBS1
+k
+.
+This simplifies the analysis of Hochschild homology in the rational setting and we can phrase a
+strong version of the HKR-theorem, which has been well known (e.g. [Qui70]). However, we would
+like to emphasize on all the structure the following result captures and give a different proof as in
+the cited source:
+Proposition 2.7. If Q ⊂ k, then for every smooth discrete k-algebra R, there are natural equival-
+ences
+HH(R/k)
+∼
+−→ UΩH
+R/k ≃
+�
+n∈N
+Ωn
+R/k[n]
+of commutative algebras in D(k) with S1-action, where the S1-action on the right hand side is given
+by the de Rham differential (cf. Construction 2.5).
+Proof. In the category CAlgBS1
+k
+Hochschild homology enjoys a universal property: For every com-
+mutative k-algebra S with S1-action any non-equivariant map R → S extends uniquely up to
+contractible choice over R → HH(R/k). Thus we get the dashed S1-equivariant algebra map
+R ≃ Ω0
+R/k
+�
+n∈N Ωn
+R/k[n]
+HH(R/k)
+The original computation in the HKR-theorem [HKR62] gives an equivalence ΩH
+R/k
+∼
+−→ HH∗(R/k)
+of differentially graded algebras. Postcomposing with the map above on homotopy groups, we get
+a map ΩH
+R/k → HH∗(R/k) → ΩH
+R/k. Finally, ΩH
+R/k has a universal property among commutative
+differentially graded algebras, as the initial CDGA with a map from R into its zeroth part. Because
+on the zeroth part the composition above is given by the identity R → R, the same is true for the
+entire map, forcing HH∗(R) → ΩH
+R/k to be an equivalence.
+□
+
+6
+KONRAD BALS
+Remark 2.8. This strong version of the HKR-theorem can be understood as a rigidification of the
+Hochschild homology functor from polynomial k-algebras Polyk: It gives a functorial factorization
+CDGAk
+Polyk
+CAlgModAD(k),
+U
+ΩH
+−/k
+HH(−/k)
+through the functor ΩH
+−/k : Polyk → CDGAk of 1-categories.
+We can now get a very good understanding of the Tate-construction for such formal objects:
+Definition 2.9. For (C∗, d) ∈ CDGAk we write |C∗| for the chain algebra (C−∗, d). This gives a
+functor |−|: CDGAk → CAlgCh∗(k) of 1-categories. More generally, given a graded object C∗ with
+differential d we want to write |C∗| to stress that we view it as a chain complex.
+Example 2.10. By definition we have |ΩH
+R/k| ≃ Ω∗
+R/k.
+With this notation set, we can make the classical computations of periodic cyclic homology in
+characteristic zero. This is also done for example in the lectures [KN18].
+Lemma 2.11. For (C∗, d) ∈ CDGAk there is a natural map |C∗| → (UC∗)tS1 in CAlgk.
+Proof. Because of the lax monoidal natural transformation (−)hS1 → (−)tS1, it suffices to estab-
+lish a natural map |C∗| → ChS1
+∗
+. Under the symmetric monoidal equivalence Fun(BS1, D(k)) ≃
+ModAD(k) the functor (−)hS1 corresponds to mapA(k, −). A choice of projective resolution P∗ of
+k as an A-coalgebra reduces us to give a functorial map |C∗| → mapA(P∗, UC∗) where the right
+hand side is the 1-categorical mapping chain complex. Now put P∗ = (A⟨t∨⟩, dP ) as the free divided
+power algebra on a primitive generator t∨ in degree 2 with dP t∨ = ǫ. Thus, computing the mapping
+chain complex gives an equivalence
+mapA(P∗, UC∗) ∼= (UC∗�t�, td)
+for |t| = −2 a dual generator to t∨ and we can explicitly describe a multiplicative chain map
+|Ci| → (UC∗�t�, td) given by Ci ≃ Ci · ti → UC∗�t� on chain groups. This finishes the proof.
+□
+Remark 2.12. The computation in Lemma 2.11 actually completely describes UChS1
+∗
+and under the
+equivalence UCtS1
+∗
+≃ UChS1
+∗
+⊗khS1 ktS1 we already get a full identification UCtS1
+∗
+≃ (UC∗((t)), td).
+For C∗ = ΩH
+R/k for R smooth over a rational algebra k we thus could have a full understanding
+of HP(R/k). However, we will not directly use this, but rather proof a general statement with more
+structure, that generalizes to non-smooth and animated algebras.
+3. Main Theorem
+Notation 3.1. Given C∗ ∈ CDGAk, we denote by Fil•
+H|C∗| the filtration
+Filn
+H|C∗| := |τ≥nC∗|
+where τ≥nC∗ is the part of grading greater or equal n. Unraveling, Fil•
+H|C∗| precisely gives the
+stupid or brutal filtration on the chain complex |C∗| ∈ Ch∗(k).
+Moreover, for X ∈ Fun(BS1, Sp) let Fil•
+T XtS1 be the Tate filtration on XtS1, see Appendix B for
+more details. It is a complete commutatively multiplicative and exhaustive filtration with associated
+graded grnFilT XtS1 ≃ X[−2n]. The Tate-filtration also restricts to a complete (and exhaustive)
+filtration on Fil0
+T XtS1 ≃ XhS1.
+
+PERIODIC CYCLIC HOMOLOGY OVER Q
+7
+Theorem 3.2. For C∗ ∈ CDGAk the map |C∗| → UCtS1
+∗
+refines and extends to an equivalence
+�
+Fil•
+H|C∗| ⊗Day Fil•
+T ktS1�∧
+→ Fil•
+T UCtS1
+∗
+of commutatively multiplicative filtered objects in D(k).
+Proof. In the concrete description of ChS1
+∗
+in Lemma 2.11, we can identify the Tate-filtration with
+the t-adic filtration on (C∗�t�, td) via Proposition B.12 and the map from Lemma 2.11 refines to
+a map of commutatively multiplicative filtrations Fil•
+H|C∗| → Fil•
+T UCtS1
+∗
+. Because the target is a
+module over the commutative algebra Fil•
+T ktS1, we get the map
+Fil•
+H|C∗| ⊗Day Fil•
+T ktS1 → Fil•
+T UCtS1
+∗
+(2)
+and because the target is complete, it even factors over the completion. To show that we get an
+equivalence of complete filtrations, it is enough to check on associated graded. Let us introduce a
+formal character t in degree −2 to visually relate Tate filtrations and t-adic filtrations and write
+grnFil•
+T ktS1 ≃ k[−2n] =: k · tn. Then on the nth associated graded the map (2) is given by
+�
+i+j=n
+Ci[−i] ⊗ k · tj ≃
+�
+i+j=n
+Ci[i] · ti ⊗ k · tj → UC∗ · tn
+and thus an equivalence by construction, as UC∗ ≃ � Ci[i] as a complex.
+□
+We can finally return to our situation of interest and immediately get a description of HP(R/k)
+in more general situations:
+Corollary 3.3. If k is an animated ring with rational homotopy groups and R in (CAlgan)k/, then
+there is an equivalence of commutatively multiplicative complete filtrations
+�
+Fil•
+HLΩ∗
+R/k ⊗Day Fil•
+T ktS1�∧
+→ Fil•
+T HP(R/k).
+(3)
+Proof. First assume that k is discrete. We want to show that both sides commute with sifted colimits
+as functors to Fil∧(D(k)). For Fil•
+HLΩ∗
+R/k after completion this is by definition and because the
+Day convolution tensor product commutes with all colimits it follows for the left hand side. As
+functors to complete filtrations we can also check this on associated graded for the right hand side:
+And also here any shifts of HH(R/k) commute with sifted colimits.
+We thus can reduce to the case that k is an ordinary Q-algebra and R smooth over k. Then the
+equivalence immediately follows from Theorem 3.2 by putting C∗ = ΩH
+R/k.
+In the general case of an animated morphism k → R between animated Q-algebras we can give
+the exact same proof. Choose a simplicial resolution kn → Rn of polynomial algebras. Again by
+definition Fil•
+HLΩ∗
+R/k ≃ colim Fil•
+HLΩ∗
+Rn/kn and thus the left hand side is determined by its value
+on polynomial rings. On the right hand side we check again, that on associated graded we get an
+equivalence
+HH(R/k) ≃ HH(R/Q) ⊗HH(k/Q) k ≃ colim HH(Rn/Q) ⊗HH(kn/Q) kn
+where the first equivalence comes from the base-change formula for Hochschild homology (cf.
+[AMN18] proof of Theorem 3.4) and the second from the facts that HH(−/Q) commutes with
+colimits in CAlgQ and that the colimit is sifted.
+Thus also in the general case, the statement
+reduces to Theorem 3.2.
+□
+Finally, in order to compute the periodic cyclic homology in our case, we only have to understand
+the left hand filtration in (3).
+There are basically two obstacles, that we have to take care of:
+Completion does not behave well with Day convolution and does not behave well with underlying
+objects.
+
+8
+KONRAD BALS
+Theorem 3.4. Let k be an animated ring with Q ⊂ π0k and X a derived scheme over k. Then
+there is a natural equivalence of underlying objects in Modk
+HP(X/k) ≃
+�
+n∈Z
+�
+LΩ∗
+X/k[−2n]
+Proof. Because both sides are sheaves in the Zariski topology on X we are reduced to the case
+X = SpecR for R ∈ CAlgan
+k/. By the above Corollary 3.3 there is a natural equivalence of filtrations
+�
+Fil•
+HLΩ∗
+R/k ⊗Day Fil•
+T ktS1�∧
+→ Fil•
+T HP(R/k).
+Because the Tate-filtration is exhaustive on HP(R/k) it suffices to compute the underlying object of
+the left filtration. Now the filtration FilT ktS1 carries a canonical splitting, because the connecting
+homomorphism in
+Filn+1
+T
+ktS1
+Filn
+T ktS1
+grnFil•
+T ktS1
+khS1[−2(n + 1)]
+khS1[−2n]
+k[−2n]
+is forced to vanish for degree reasons, in fact Map(k[−2], khS1[−2n − 3]) is contractible. Therefore,
+we have a map of filtrations �
+n≥• k[−2n] → Fil•
+T ktS1, inducing an equivalence on associated graded,
+and, thus, as the left hand side is complete, it even is an equivalence of filtrations.
+We claim now, that this splitting induces an equivalence
+�
+n∈Z
+(Fil•−n
+H
+LΩ∗
+R/k[−2n])∧ ≃ (FilHLΩ∗
+R/k ⊗Day FilT ktS1)∧
+Indeed, the canonical map �
+n∈Z(Fil•−n
+H
+LΩ∗
+R/k[−2n]) → �
+n∈Z(Fil•−n
+H
+LΩ∗
+R/k[−2n])∧ exhibits the
+right hand side as the completion: It is evidently complete and the map on the m-th associated
+graded
+�
+n∈Z
+LΩm−n
+R/k [−2n] →
+�
+i∈Z
+LΩm−n
+R/k [−2n]
+is an equivalence, because LΩm−n
+R/k is always bounded below and 0 for n > m.
+Finally we want to compute the underlying object, i.e.
+the colimit.
+Consider the canonical
+colimit-limit-interchange can map sitting in the cofibre sequence
+colim
+��
+n∈Z
+Fil•−n
+H
+�
+LΩ∗
+R/k[−2n]
+�
+can
+−−→
+�
+n∈Z
+�
+LΩ∗
+R/k[−2n] → colim
+��
+n∈Z
+LΩ≤•−n−1
+R/k
+[−2n]
+�
+But because LΩ≤•−n−1
+R/k
+is bounded below for all n, and 0 for n ≥ •, the right most product is
+actually degreewise finite and, thus, vanishes in the colimit. Now putting everything together gives
+the result.
+□
+We want to use the result to investigate the exhaustiveness of the HKR-filtration constructed in
+[Ant19]. It arises from the left Kan extension of the Beilinson Whitehead tower of the Tate filtration
+on HP(−/k) from smooth algebras to bicomplete filtrations as the underlying outer filtration. For
+more details c.f. loc. cit. or [BL22] section 6.3.
+Proposition 3.5. In the situation of the Theorem 3.4, the HKR-filtration on HP(R/k) can be
+identified with the filtration by partial products of �
+n∈Z
+�
+LΩ∗
+Rn/kn[−2n]. Precisely,
+Fili
+HKR HP(R/k) ≃
+�
+n≤−i
+�
+LΩ∗
+Rn/kn[−2n]
+
+PERIODIC CYCLIC HOMOLOGY OVER Q
+9
+Proof. By definition of the HKR-filtration we only have to construct equivalences in the case R over
+k a smooth algebra. But now the Tate-filtration on HP(R/k) induces a shifted Hodge filtration on
+the factor Ω∗
+R/k[2n] with Film
+T (Ω∗
+R/k[2n]) ≃ (Filn+m
+H
+Ω∗
+R/k)[2n]. Because Filn+m
+H
+Ω∗
+R/k ∈ D(k)≤−n−m
+we have
+Film
+T (Ω∗
+R/k[2n]) ∈ D(k)≤n−m
+Moreover, we can similarly compute
+grmFil•
+T (Ω∗
+R/k[2n]) ≃ Ωn+m
+R/k [−n − m + 2n] ∈ D(k)≥n−m.
+In fact these two conditions precisely show that Fil•
+T (Ω∗
+R/k[2n]) is concentrated in degree n with
+respect to the Beilinson t-structure on Fil(D(k)). From our complete description of Fil•
+T HP(R/k)
+in terms of Ω∗
+R/k · [2n] we get
+Fil•
+T (Ω∗
+R/k[2n]) ≃ πnFil•
+T HP(R/k) ≃ grnFil•
+HKR HP(R/k)
+where the last equivalence comes form the definition of the HKR-filtration. In particular, HP(R/k)
+decomposes into the product of the associated gradeds of the HKR-filtration, which proves the
+claim.
+□
+Corollary 3.6. In the situation of the theorem the HKR-filtration from [Ant19] is exhaustive if and
+only if �
+LΩ∗
+X/k is bounded above.
+Proof. We can phrase the exhaustiveness as the condition that the natural map
+colim
+i
+�
+n≥i
+�
+LΩ∗
+X/k[2n] →
+�
+n∈Z
+�
+LΩ∗
+X/k[2n] ≃
+�
+n∈Z
+�
+LΩ∗
+X/k[−2n]
+is an equivalence. This is precisely the case when �
+LΩ∗
+X/k[2n] eventually leaves any fixed degree for
+n → ∞, precisely when it is bounded above.
+□
+Example 3.7. In [Ant19] Antieau proves without assumptions on the discrete commutative base ring
+k, that the HKR-filtration is exhaustive if X is quasi-lci over k, i.e. LΩ1
+R/k has Tor-amplitude in
+[0, 1]. We recover this statement in our situation via the observation that the lci-condition forces
+�
+LΩ∗
+X/k to be concentrated in degrees (−∞, 0].
+Moreover, with a result in [Bha12] in the rational setting we can even prove a more drastic result:
+Corollary 3.8. If k is a discrete Q-algebra and X a qcqs-scheme over k, then the HKR-filtration
+is exhaustive.
+Proof. By the last Corollary we want to prove that �
+LΩ∗
+X/k is bounded above. Because X is qcqs
+and �
+LΩ∗
+−/k is a sheaf, its global sections on X are computed by a finite limit of the value on affines
+(cf. Remark A.7). Thus, it satisfies to show the claim for X = SpecR with an arbitrary k-algebra
+R. If we write k → R as a filtered colimit of maps (kn → Rn)n∈N in CAlg(D(k0)♥)∆1, where kn is
+Noetherian and Rn is of finite type over kn, we get �
+LΩ∗
+R/k ≃ colim
+�
+LΩ∗
+Rn/kn in D(k0). Hence, we
+can further reduce the claim to the case k Noetherian and R finite type over k. In this situation
+the result of Theorem 4.10 in [Bha12] gives a concrete description of �
+LΩ∗
+R/k, in particular it sits in
+homological degree (−∞, 0].
+□
+4. Multiplicative Structure
+In the Corollary 3.3 the equivalence
+�
+Fil•
+HLΩ∗
+R/k ⊗Day Fil•
+T ktS1�∧
+→ Fil•
+T HP(R/k)
+
+10
+KONRAD BALS
+was compatible with the commutative algebra structures on both sides. Thus we are able to deduce
+properties of the induced commutative algebra structure on �
+n∈Z �
+LΩ∗
+R/k[−2n]. But first we will
+describe algebra structures on these big products more generally:
+Definition 4.1. Given a complete and exhaustive commutative multiplicative filtration R• ∈
+CAlgFil(Modk) on a commutative algebra R ∈ CAlgk. We define
+R�t±1� := colim
+�
+R• ⊗Day Fil•
+T ktS1�∧
+Example 4.2. If R ∈ CAlgk for an animated commutative ring k, equipped with the constant
+negatively graded filtration, then we have R�t±1� ≃ RtS1 with respect to the trivial S1-action on R.
+If moreover, π0k is rational, we can even write R((t)) := RtS1 as the unique commutative algebra in
+Modk with homotopy groups π∗R((t)) for a generator |t| = −2.
+Corollary 4.3. In the situation of Theorem 3.4 the equivalence refines to a natural equivalence
+HP(X/k) ≃ �
+LΩ∗
+X/k�t±1� in CAlgk.
+In fact, in the situation of Corollary 4.3 we can demystify the object �
+LΩ∗
+X/k�t±1�. The object
+R�t±1� does not fully depend on R• as a complete filtered object. We will show a very special case,
+of this feature:
+Lemma 4.4. If F • ∈ Modk is a filtered object with F n = 0 for n but finite n, then colim(F • ⊗Day
+Fil•
+T ktS1)∧ ≃ 0. In particular, if R• → R• is a map in CAlgFil(Modk)∧ such that the maps induce
+equivalences for all but finite n, then R�t±1� ≃ R�t±1�.
+Proof. As in the proof of Theorem 3.4, we get an equivalence �
+n∈Z F •−n[−2n]
+∼
+−→ F •⊗DayFil•
+T ktS1.
+However, the left hand side is already complete: F •−n is complete because it is eventually 0 and
+the direct sum is in fact a product, because there are only finitely many non-zeros factors. Finally,
+the underlying object of F • is 0 and thus also of the complete filtration F • ⊗Day Fil•
+T ktS1.
+For the last statement, we note that the construction colim(− ⊗Day Fil•
+T ktS1)∧ is exact.
+□
+Proposition 4.5. In Corollary 4.3 we can have further identifications of commutative algebras
+HP(X/k) ≃ �
+LΩ∗
+X/k�t±1�
+∼
+−→ limm LΩ≤m
+X/k((t)).
+Proof. We start in a general setting: Given a complete multiplicative exhaustive filtration R• on a
+k-algebra R. Set R•/Rm to be the filtration with (R•/Rm)(l) := Rl/Rm for l ≤ m and 0 otherwise.
+Because R• is complete we have R•
+∼
+−→ limm R•/Rm. Checking on associated graded we get an
+equivalence
+�
+R• ⊗Day Fil•
+T ktS1�∧
+∼
+−→ lim
+m
+�
+R•/Rm ⊗Day Fil•
+T ktS1�∧
+and thus the natural map R�t±1� → limm(R/Rm�t±1�). Now if R• is eventually constant in negative
+degrees, and because it is eventually 0 in positive degrees R•/Rm�t±1� ≃ R/Rm((t)) by Lemma 4.4
+and Example 4.2.
+Finally, in the concrete situation R• = Fil•
+H �
+LΩ∗
+X/k, which satisfies this last assumption, we have
+an easy description of the quotients �
+LΩ∗
+X/k/ �
+LΩ≥m+1
+X/k
+≃ LΩ≤m
+X/k. And now the proof of Theorem
+3.4 gives an equivalence LΩ≤m
+X/k�t±1� ≃ �
+n∈Z LΩ≤m
+X/k[−2n] on underlying objects, such that the
+map from �
+LΩ∗
+X/k�t±1� can be identified with the natural map �
+LΩ∗
+X/k → LΩ≤m
+X/k in each factor. In
+particular this map is an equivalence in the limit.
+□
+We can finally get to the description of the homotopy groups HP∗(X/k) explained in the in-
+troduction. Disregarding the multiplicative structure on HP∗(X/k) Theorem 3.4 already gives the
+
+PERIODIC CYCLIC HOMOLOGY OVER Q
+11
+additive identification
+HP∗(X/k) ∼=
+�
+n∈Z
+π∗+2n �
+LΩ∗
+X/k ∼=
+��
+n∈Z
+antn : an ∈ π∗+2n �
+LΩ∗
+X/k
+�
+with the componentwise addition as stated in the introduction. We will now show how to describe
+the multiplication: Given
+��
+n∈Z antn�
+,
+��
+n∈Z bntn�
+∈ HP∗(X/k), then we know that
+��
+n∈Z
+antn
+�
+·
+��
+n∈Z
+bntn
+�
+=
+�
+n∈Z
+cntn
+(4)
+for some cn ∈ π∗ �
+LΩ∗
+X/k, so that we want to describe these coefficients cn.
+Construction 4.6. The graded ring π∗ �
+LΩ∗
+X/k can be equipped with the coarsest topology making
+all maps π∗ �
+LΩ∗
+X/k → π∗LΩ≤m
+X/k continuous for the discrete topology on the target. Concretely, this
+means a neighborhood basis of 0 is given by the kernels of these maps above. In particular, the
+topology cannot separate points that lie in every single such kernel, i.e. lie in the kernel of the
+surjective map π∗ �
+LΩ∗
+X/k → lim π∗LΩ≤m
+X/k. In degree i this is precisely given by lim1 πi+1LΩ≤m
+X/k. In
+fact lim π∗LΩ≤m
+X/k is the "Hausdorffization" of this non-Hausdorff topology.
+Lemma 4.7. In the equation (4) the coefficient cn is a limit of the net �
+i+j=n ai · bj.
+Proof. It is enough to prove this statement for homogeneous elements, and for simplicity assume
+that (�
+n∈Z antn) and (�
+n∈Z bntn) are both in degree 0. For the general case, one only has to
+correctly modify the degrees of elements, the arguments are the same.
+By definition of the topology on π∗ �
+LΩ∗
+X/k we have to show, that cn − �
+(i,j)∈Jn ai · bj for finite
+Jn ⊂ {i + j = n} eventually lies in the kernel of the maps π∗ �
+LΩ∗
+X/k → π∗LΩ≤m
+X/k. By Proposition
+4.5 these maps assemble to ring maps
+ϕn : HP∗(X/k) → π∗LΩ≤m
+X/k((t)),
+where we understand the multiplication of Laurent-series on the target. Moreover, because the
+coefficients of the target are in degrees ≥ −m as a graded ring, we even know, that ai · bj is sent
+to 0 in π∗LΩ≤m
+R/k as soon as i < m/2 or j < m/2.
+That means for every family of finite sets
+Jn ⊂ {i + j = n} containing In := {i + j = n : i, j ≥ m/2}
+ϕn
+��
+n∈Z
+��
+Jn
+ai · bj
+�
+tn
+�
+=
+�
+n∈Z
+��
+In
+ϕn(ai) · ϕn(bj)
+�
+tn
+=
+��
+n∈Z
+ϕn(an)tn
+�
+·
+��
+n∈Z
+ϕn(bn)tn
+�
+But also by definition we have ϕn
+��
+n∈Z cntn�
+=
+��
+n∈Z ϕn(an)tn�
+·
+��
+n∈Z ϕn(bn)tn�
+. In particular,
+taking the difference and restricting again to single coefficients cn − �
+Jn ai · bj is sent to 0 in
+π∗LΩ≤m
+X/k.
+□
+This concludes the description of HP∗(X/k) given in the introduction.
+Appendix A. HP of Schemes
+In this section, we want to carefully describe the extension of Hochschild and periodic cyclic
+homology to (derived) schemes. We will refer to [Lur18] [Section 1.1], [Lur10] and [Toë14] for an
+introduction to derived schemes over animated commutative (aka simplicially commutative) rings.
+We will only sketch the definition:
+
+12
+KONRAD BALS
+Definition A.1. For an animated commutative k-algebra R, define the affine derived scheme SpecR
+to be the pair (|SpecR|, OSpecR) where |SpecR| = |Specπ0R| is a topological space and OSpecR is
+a CAlgan
+k/-valued sheaf on |SpecR| with OSpecR(D(f)) ≃ R[f −1] for every elementary open D(f) ⊂
+|Specπ0R|.5
+A general pair X = (|X|, OX) with |X| a topological space and OX ∈ ShvCAlgan
+k/(|X|) is called a
+derived scheme, if there exist an open cover U of X, such that for all U ∈ U we have (U, OX|U) ∼=
+SpecR6 for some R ∈ CAlgan
+k/.
+Remark A.2. This notion generalizes ordinary schemes. In particular given a derived scheme X, the
+underlying ringed space π0X := (|X|, π0OX) is an ordinary scheme and we call a derived scheme X
+affine7, quasi-affine, quasi-compact resp. quasi-separated if π0X is so.
+Definition A.3. Let X be a derived k-scheme. A Zariski-sheaf with values in a category C on X
+is a C-valued sheaf on the topological space |X|, i.e. a functor F : U(X)op → C from the opposite of
+the poset U(X) of opens of |X|, satisfying
+F(U) ≃
+lim
+∅̸=S⊂I
+finite
+F(US)
+for every U = �
+i∈I Ui ∈ U(X) and with US = Ui0 ∩ . . . ∩ Uik for S = {i0, . . . , ik}.
+Given a derived scheme X over k the goal is now to upgrade the functors HH(−/k), HP(−/k):
+CAlgan
+k/ → Modk to Zariski-sheaves HHk and HPk on X in order to define HH(X/k) := Γ(X, HHk)
+and HP(X) := Γ(X, HPk).
+Proposition A.4. Given a topological space X and Ue a set of open subsets of X, such that
+1) Ue forms a basis of the topology of X,
+2) Ue is closed under intersections.
+Then the adjunction
+Fun(U(X)op, C)
+Fun(Uop
+e , C)
+res
+Ran
+restrict to an equivalence of sheaf cat-
+egories ShvC(X)
+∼
+−→ ShvC(Ue) with the induced Grothendieck topology on Ue.
+If, moreover, Ue
+consist of quasi-compact opens, then ShvC(X) ≃ Fun′(Uop
+e , C), where the right hand side consists
+of those presheaves F : Uop
+e
+→ C, that satisfy F(∅) = 0 and F(U ∪ V ) ≃ F(U) ×F(U∩V ) F(V ) for
+U, V, U ∪ V ∈ Ue.
+Proof. The first statement is a special case of the infinity categorical comparison Lemma for Grothen-
+dieck sites proven in [Hoy14] Lemma C.3, and the second claim is [Lur18] Proposition 1.1.4.4.
+□
+We now do the standard procedure of extending an algebraic functor CAlgan
+k/ → C to a sheaf on
+geometric objects. We proceed in steps:
+Lemma A.5. Given a quasi-affine derived scheme X over k, there are Modk-valued sheaves HHk
+and HPk on X, extending HH(−/k) and HP(−/k), i.e.
+for all affine open derived subschemes
+U ⊂ X, the sheaves recover Hochschild homology, resp. periodic cyclic homology:
+Γ(U, HHk) ≃ HH(OX(U)/k)
+Γ(U, HPk) ≃ HP(OX(U)/k)
+Proof. Set Ue to be the set of affine open derived subschemes of X. Then HH(−/k) and HP(−/k) give
+functors Uop
+e
+→ Modk and let HHk and HPk denote their right Kan extension along Uop
+e
+→ U(X)op.
+We want to argue, that these are already Zariski-sheaves on X.
+Because X is quasi-affine, intersections of affines are computed in a surrounding affine derived
+scheme, and are affine again. The collection Ue, thus, satisfies the conditions 1), 2) of Propos-
+ition A.4 and contains only quasi-compact opens, so that we are reduced to checking that the
+5The existence of SpecR is deduced in [Lur18] from Proposition A.4 below.
+6Under the appropriate notion of equivalence.
+7In fact X is affine, if and only if X = SpecR for R ∈ CAlgan
+k/.
+
+PERIODIC CYCLIC HOMOLOGY OVER Q
+13
+functors HH(−/k) and HP(−/k) satisfy the finite limit condition of Fun′(Uop
+e , Modk). As the Tate-
+construction commutes with finite limits, it is enough to only show the claim for Hochschild homo-
+logy.
+For R ∈ CAlgan
+k/ the natural map R → HH(R/k) in CAlgk equips HH(R/k) with a module
+structure over R, such that for a map of animated commutative rings R → R′ the functoriality
+induces a map HH(R/k) ⊗R R′ → HH(R′/k) in ModR′. Now if U ⊂ X is an affine open derived
+subscheme of X, then for every other affine open V ⊂ U this map
+HH(OX(U)/k) ⊗OX(U) OX(V ) → HH(OX(V )/k)
+is an equivalence. Indeed, it suffices to check this locally on V , so we can reduce to distinguished
+opens D(f) ⊂ V ⊂ U for f ∈ π0OX(U). But using that HH(−/k) commutes with filtered colimits
+we can identify both sides with HH(OX(U)[f −1]).
+Finally, assume that F : I → Ue is a finite diagram with colimit U as appearing in Proposition
+A.4, then by the above HH(OX(F op(−))/k) ≃ HH(OX(U)/k) ⊗OX(U) OX(F op(−)) and we win as
+tensoring is exact and OX(F op(−)) is a finite limit diagram due to the sheaf condition of OX (using
+Proposition A.4 in the other direction).
+□
+Lemma A.6. Given an arbitrary derived k-scheme X, we can furthermore extend Hochschild and
+periodic cyclic homology to sheaves HHk and HPk on X.
+Moreover, for all open qcqs derived
+subschemes U ⊂ X we have
+Γ(U, HPk) ≃ Γ(U, HHk)tS1
+Proof. Let Ue now be the set of quasi-affine open derived subschemes of X, which satisfies 1) and
+2) of Proposition A.4. By the last Lemma HH(−/k) and HP(−/k) extend to sheaves on Uop
+e
+and
+by Proposition A.4 thus further extend to sheaves on entire X.
+Now take U ⊂ X a quasi-compact quasi-separated derived open subscheme. Because of quasi-
+compactness there exist a finite open cover of U by affine open subschemes U1, . . . Un and by the
+sheaf condition we get
+Γ(U, HPk) ≃
+lim
+S⊂[1,n] Γ(US, HPk)
+in the notation of Definition A.3. Each US is now quasi-affine as an open derived subscheme of an
+affine and quasi-compact by the quasi-separatedness of U. Thus, because the limit above is finite, it
+satisfies to check the claim for U quasi-compact quasi-affine. Again, choosing a finite open cover by
+affines and using that the intersection of affines in quasi-affines is affine again, we can even reduce
+to the case that U is an affine open. But in this case
+Γ(U, HPk) ≃ HP(OX(U)/k) ≃ HH(OX(U)/k)tS1 ≃ Γ(U, HHk)tS1
+□
+Remark A.7. The proof of the last Lemma shows even more: For any sheaf F on a derived scheme
+X, the sections Γ(U, F) over a qcqs open derived subscheme U are computed as a finite limit of the
+values of F on affines.
+Appendix B. Tate Filtration
+In this section we want to review the construction of the classical Tate-filtration introduced in
+[GM95]. This content is not new and also recently has been explained in [BL22] section 6.1. We
+would like to particular put a focus on multiplicative structures.
+Definition B.1. Given a representation ρ: S1 → GL(V ) of S1, the representation sphere SV is the
+one-point compactification of V . Furthermore we define SV := Σ∞SV as the suspension spectrum
+of the representation sphere.
+Remark B.2. Note that if V is finite dimensional there immediately is an equivalence SV ≃ SdimR V ,
+so that the homotopy type of SV only depends on the dimension of V . However, the S1-action
+really uses the representation S1 → GL(V ).
+
+14
+KONRAD BALS
+Example B.3. For V = C there is the standard representation given by S1 ≃ U(1) ֒→ C×. Its
+representation sphere sits in the pushout
+S1
+∗
+∗
+SV
+with S1-acting freely on itself.
+Thus, after adding basepoints to the top row Σ∞ gives a fibre
+sequence S[S1] := Σ∞
++ S1 → S → SV of spectra with S1-action.
+Construction B.4. Let V be a finite dimensional representation of S1. The map 0 → V of repres-
+entations induces a sequence
+0 → V → V ⊕ V → V ⊕ V ⊕ V → · · ·
+which translates to the representation sphere spectra to a Z-graded filtration
+S•V := · · · → S−2V → S−V → S → SV → S2V → S3V → · · ·
+(5)
+where S−nV := DSnV is the Spanier-Whitehead dual. Now if V ̸= 0 all maps have to be non-
+equivariantly nullhomotopic, but this is definitely not the case with respect to the S1-action. We
+will see this later in Proposition B.6.
+Definition B.5. Given a spectrum X ∈ SpBS1 with S1-action, we define the Tate-filtration
+FilT XtS1 as
+· · · →
+�
+S−2V ⊗X
+�hS1
+→
+�
+S−V ⊗X
+�hS1
+→XhS1→
+�
+SV ⊗ X
+�hS1
+→
+�
+S2V ⊗X
+�hS1
+→ · · ·
+for V the standard representation of S1 constructed in Example B.3.
+This definition would not be sensible if this would not give a filtration on XtS1 and we are bound
+to prove:
+Proposition B.6. For X ∈ SpBS1 the Tate filtration Fil•
+T XtS1 is complete with underlying object
+XtS1.
+Proof. Because homotopy fixed points, as a limit, preserve completeness it satisfies to prove that
+lim S−nV ⊗ X ≃ 0 as a spectrum. But here we can compute
+lim S−nV ⊗ X ≃ lim map
+�
+SnV , X
+�
+≃ map
+�
+colim SnV , X
+�
+≃ 0
+because the colimit goes along nullhomotopic maps. However, as already indicated, those maps are
+not equivariantly nullhomotopic In fact every map
+S−nV ⊗ X → S(−n+1)V ⊗ X
+induces an equivalence on the S1-Tate construction. Indeed, via Example B.3 we can identify the
+fibre as S−nV ⊗ S[S1], which is an induced S1-spectrum, such that Tate vanishes on this fibre. Now
+we can look at the Z-indexed fibre sequences defining the Tate constructions of S−nV ⊗ X:
+Σ
+�
+S−nV ⊗ X
+�
+hS1
+�
+S−nV ⊗ X
+�hS1
+�
+��
+�
+≃Filn
+T XtS1
+�
+S−nV ⊗ X
+�tS1
+.
+By the observation above the right hand filtration is constant at XtS1. The colimit of the left hand
+filtration vanishes, because commuting the colimit with Σ(− ⊗ X)hS1 reduces again to computing
+a filtered colimit along nullhomotopic maps, which is 0. Thus together we see colim Filn
+T XtS1 ≃
+XtS1.
+□
+
+PERIODIC CYCLIC HOMOLOGY OVER Q
+15
+We are interested in possible algebra structures on the Tate filtration. Because (−)tS1 is lax
+monoidal, XtS1 for an algebra X ∈ Alg(Sp) inherits an algebra structure again.
+However, the
+question of algebra structures on FilT XtS1 with respect to the Day convolution is more subtle.
+We will use the following different description of the filtered category as a modules over a graded
+algebra. This insight comes from Lurie in [Lur15] 3.2 and in this form is in [Rak20] Proposition
+3.2.9.
+Definition B.7. For a stable symmetric monoidal category C with unit
+1, let
+1[β] denote the
+underlying graded object of the unit in Fil(C). It is a commutative algebra in Gr(C) with underlying
+graded object �
+n≤0
+1.
+Every object in the symmetric monoidal category Fil(C) is canonically a module over the unit,
+such that the symmetric monoidal forgetful functor Fil(C) → Gr(C) refines to a functor Fil(C) →
+Mod
+1[β](Gr(C)). In fact remembering this action of
+1[β] recovers the full filtered object:
+Theorem B.8 ([Rak20]). For a symmetric monoidal stable category C the above functor Fil(C) →
+Mod
+1[β](Gr(C)) is an equivalence of symmetric monoidal categories.
+In our situation we want to use this for C = SpBS1
+Q
+and show that the filtered object S−•V ⊗ Q is
+a commutative algebra in Fil(SpBS1
+Q
+). There is also an algebraic description of this category due to
+Greenlees-Shipley [GS09] in the non-Borel-complete setting and later as we use it here by [MNN17].
+Similar to above, because again in SpQ every object carries a canonical module structure over the
+unit Q, the lax functor (−)hS1 : SpBS1
+Q
+→ SpQ refines to a functor into ModQhS1 (SpQ) and we have
+as a special case of Theorem 7.35 in [MNN17]:
+Theorem B.9 ([MNN17]). The functor (−)hS1 : SpBS1
+Q
+→ ModQhS1 (SpQ) is fully faithful with
+essential image given by those modules over QhS1 ≃ Q�t� that are complete with respect to the t-adic
+filtration.
+The Thom isomorphism over Q for complex vector bundles over BS1 gives an S1-equivariant
+equivalence of SV ⊗Q ≃ Q[2] with trivial S1-action on the right. Thus the map S⊗Q → SV ⊗Q ≃ Q[2]
+in SpBS1
+Q
+corresponds to an Q�t�-module map Q�t� → Q�t�[2] for |t| = −2. In particular as a Q�t�-
+module map it is determined by the image of 1 in Q·t. Because this map is not 0 as seen in the proof
+of Proposition B.6, up to a unit, it is given by multiplication by t. More generally this argument
+gives an identification of the image of the filtration S−•V ⊗ Q under (−)hS1 with the filtration
+· · ·
+·t−→ Q�t�[−2n]
+·t−→ Q�t�
+·t−→ Q�t�[2n]
+·t−→ · · ·
+of Q�t�-modules.
+Lemma B.10. The filtration S−•V ⊗Q can be given a commutative algebra structure in Fil(SpBS1
+Q
+).
+Proof. Under the symmetric monoidal equivalences from the cited Theorems B.8 and B.9 we are
+reduced to equip the underlying graded object �
+n∈Z Q�t�[−2n] of (S−•V ⊗Q)hS1 with a commutative
+algebra structure over Q�t�[β]. To avoid confusion, let us introduce a formal variable s in grading
+degree −1 and homological degree 2 to get an identification of underlying objects
+�
+n∈Z
+Q�t�[−2n] ≃ Q�t�[s±1],
+which is the free graded commutative Q�t�-algebra on the variables s±1. In particular sending β to
+s · t gives Q�t�[s±1] the desired commutative algebra structure.
+□
+Proposition B.11. For a commutative ring k with Q ⊂ π0k and R ∈ CAlgBS1
+k
+the filtration
+FilT RtS1 permits the structure of a commutative algebra in Fil(Modk).
+
+16
+KONRAD BALS
+Proof. By construction FilT (−)tS1 is the composite
+ModBS1
+k
+(−)⊗(S−•V ⊗k)
+−−−−−−−−−−→ Fil(ModBS1
+k
+)
+(−)hS1
+−−−−→ Fil(Modk).
+The second functor has a canonical lax structure. Because k is a commutative algebra over Q, also
+(S−•V ⊗ k) inherits a commutative algebra structure via Lemma B.10 and thus FilT (−)tS1 refines
+to a lax symmetric monoidal functor and sends commutative algebras in ModBS1
+k
+to commutative
+algebras in Fil(Modk).
+□
+Given C∗ ∈ CDGAQ
+ι֒−→ SpBS1
+Q
+we would like to conclude this section with the comparison of the
+induced filtration on Fil≥0
+T UCtS1
+∗
+on the zeroth part Fil0
+T UCtS1
+∗
+≃ UChS1
+∗
+to concrete filtrations on
+the chain level.
+Proposition B.12. In the notation of Lemma 2.11, there is an identification of the t-adic filtration
+on UChS1
+∗
+≃ (UC∗�t�, td) with the Tate filtration Fil≥0
+T UCtS1
+∗
+.
+Proof. Using the cocommutative bialgebra A := Q[ǫ]/ǫ2 as defined in Construction 2.1 we have the
+symmetric monoidal equivalence of categories SpBS1
+Q
+∼
+−→ ModASpQ (Corollary 2.3). Therefore the
+filtration Fil≥0
+T UCtS1
+∗
+reads as
+· · · → mapA(Q, Q[−4] ⊗ C∗) → mapA(Q, Q[−2] ⊗ C∗) → mapA(Q, Q ⊗ C∗) ≃ UChS1
+∗
+By duality this filtration is equivalently induced by the maps Q[2n] → Q[2n + 1] from S−•V ⊗ Q for
+n ≥ 0 in the first variable of the mapping spectrum. Choosing P∗ = (A⟨t∨⟩, dP ) for t∨ primitive in
+degree 2 and dP (t∨) = ǫ as in the proof of Lemma 2.11, these maps
+P∗[2n]
+· · ·
+k · (t∨)2
+k · ǫt∨
+k · t∨
+k · ǫ
+k
+P∗[2n + 1]
+· · ·
+k · t∨
+k · ǫ
+k
+∼
+0
+∼
+0
+∼
+0
+are uniquely determined as A-module maps by the image of t∨. Because again the map is non-zero,
+the image of t∨ has to be a unit. In particular, up to isomorphism the maps
+ChS1
+∗
+[−2n − 2] → ChS1
+∗
+[−2n]
+are given by multiplication with the dual t in (C∗�t�, td).
+□
+References
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+WWU Münster, Mathematisches Institut, Einsteinstr. 62, 48149 Münster, Germany
+Email address: [email protected]
+

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+CaSpeR: Latent Spectral Regularization for Continual Learning
+Emanuele Frascaroli1,3, Riccardo Benaglia1,3, Matteo Boschini1, Luca Moschella2
+Cosimo Fiorini3, Emanuele Rodol`a2, Simone Calderara1
+1AImageLab University of Modena and Reggio Emilia
+2Sapienza University of Rome
+3Ammagamma
+Abstract
+While biological intelligence grows organically
+as new knowledge is gathered throughout life,
+Artificial Neural Networks forget catastrophi-
+cally whenever they face a changing training data
+distribution. Rehearsal-based Continual Learn-
+ing (CL) approaches have been established as a
+versatile and reliable solution to overcome this
+limitation; however, sudden input disruptions
+and memory constraints are known to alter the
+consistency of their predictions. We study this
+phenomenon by investigating the geometric char-
+acteristics of the learner’s latent space and find
+that replayed data points of different classes in-
+creasingly mix up, interfering with classification.
+Hence, we propose a geometric regularizer that
+enforces weak requirements on the Laplacian
+spectrum of the latent space, promoting a par-
+titioning behavior. We show that our proposal,
+called Continual Spectral Regularizer (CaSpeR),
+can be easily combined with any rehearsal-based
+CL approach and improves the performance of
+SOTA methods on standard benchmarks.
+Fi-
+nally, we conduct additional analysis to provide
+insights into CaSpeR’s effects and applicability.
+1
+INTRODUCTION
+Within the natural world, intelligent creatures continually
+learn to adapt their behavior to changing external condi-
+tions. In doing so, they seamlessly blend novel notions
+with previous understanding into a cohesive body of knowl-
+edge. On the contrary, ANNs greedily fit the data they are
+currently trained on. For a model that learns on a chang-
+ing stream of data, this results in the swift deterioration of
+previously acquired information – a phenomenon known as
+catastrophic forgetting (McCloskey and Cohen, 1989).
+Continual Learning (CL) is a branch of machine learning
+that designs approaches to help deep models retain pre-
+vious knowledge while training on new data (De Lange
+et al., 2021; Parisi et al., 2019). The evaluation of these
+B
+Memory Buffer
+Feature
+Extractor
+Fθ
+B
+Memory Buffer
+Feature
+Extractor
+Fθ
+Latent Space
+Rehearsal method + C a S p e R
+Basic Rehearsal method
+Latent Graph Spectrum
+Latent Space
+λ1 λ2 λ3 λ4
+λ5 λ6
+λ7 λ8
+Eigengap
+λ2
+Eigengap
+λ3 λ4
+λ5
+λ6 λ7 λ8
+λ1
+Latent Graph Spectrum
+Figure 1: An overview of the proposed CaSpeR regular-
+izer. Rehearsal-based CL methods struggle to separate the
+latent-space projections of replay data points. Our proposal
+acts on the spectrum of the latent geometry graph to induce
+a partitioning behavior by maximizing the eigengap for the
+number of seen classes (best seen in color).
+methods is typically conducted by dividing a classification
+dataset into disjoint subsets of classes, called tasks, letting
+the model fit one task at a time and, finally, evaluating it on
+all previously seen classes (van de Ven and Tolias, 2018).
+Recent literature favors the employment of rehearsal meth-
+ods; namely, CL approaches that address forgetting by re-
+taining a small memory buffer of samples encountered in
+previous tasks and interleaving them with current training
+data (Chaudhry et al., 2019; Buzzega et al., 2020a).
+On the one hand, rehearsal is a straightforward solution that
+allows the learner to keep track of the joint distribution of
+all input classes seen so far. On the other hand, the mem-
+ory buffer can only accommodate a limited amount of past
+examples, resulting in overfitting issues (high accuracy on
+the memory buffer, low accuracy on the test set of the orig-
+inal tasks). Recent studies characterize this phenomenon
+arXiv:2301.03345v1  [cs.LG]  9 Jan 2023
+
+CaSpeR: Latent Spectral Regularization for Continual Learning
+in terms of abruptly divergent gradients upon introducing
+new classes (Caccia et al., 2022; Boschini et al., 2022b) or
+deteriorating decision surface (Bonicelli et al., 2022). A
+well-known outcome is the accumulation of a predictive
+bias in favor of the currently seen classes (Wu et al., 2019;
+Ahn et al., 2021).
+While these works focus their analysis on the overall pre-
+diction of the model, we instead consider the changes
+occurring in its latent space as tasks progress.
+Specifi-
+cally, we observe that the learner struggles to separate la-
+tent projections of replay examples belonging to different
+classes. This constitutes a weak spot for the learner, mak-
+ing the downstream classifier prone to interference when-
+ever the input distribution changes and representations are
+perturbed. Given the Riemannian nature of the latent space
+of DNNs (Arvanitidis et al., 2018), we naturally revert to
+spectral geometry to model and constrain its evolution.
+Spectral geometry is preferred over other geometric tools
+as it focuses on the latent-space structure without imposing
+constraints on individual coordinates.
+In this work, we introduce a loss term aimed at endow-
+ing the model’s latent space with a cohesive structure. Our
+proposed approach, called Continual Spectral Regularizer
+(CaSpeR), leverages graph-spectral theory to promote the
+generation of well-separated latent embeddings, as illus-
+trated Fig. 1. We show that our proposal can be seam-
+lessly combined with any rehearsal-based CL method to
+improve its classification accuracy and robustness against
+catastrophic forgetting. Moreover, since CaSpeR does not
+rely on the availability of annotations for each example, we
+show that it can be easily applied to semi-supervised sce-
+narios to provide better accuracy and easier convergence.
+In summary, we make the following contributions:
+• We study the interference in rehearsal CL models by
+investigating the geometry of their latent space. To
+the best of our knowledge, this is the first attempt at a
+geometric characterization of catastrophic forgetting;
+• We propose Continual Spectral Regularizer: a simple
+geometrically motivated loss term, inducing the online
+learner to produce well-organized latent embeddings;
+• We validate our proposal by combining it with sev-
+eral SOTA rehearsal-based CL approaches. Our re-
+sults show that CaSpeR is effective both in the Class-
+Incremental and Task-Incremental CL setting (van de
+Ven and Tolias, 2018) by increasing the geometric
+consistency of the latent space;
+• Finally, we show that CaSpeR can be beneficially
+applied also to the challenging Continual Semi-
+Supervised Learning (CSSL) scenario, producing
+higher accuracy and easier convergence.
+2
+RELATED WORK
+2.1
+Continual Learning
+Continual Learning approaches are designed to comple-
+ment and assist in-training deep learning models to mini-
+mize the incidence of catastrophic forgetting (McCloskey
+and Cohen, 1989) when learning on a changing input distri-
+bution. This aim can be pursued through different classes
+of solutions (De Lange et al., 2021): architectural meth-
+ods explicitly allocate separate portions of the model to
+separate tasks (Mallya and Lazebnik, 2018; Serra et al.,
+2018); regularization methods rely on a loss term to pre-
+vent the model from changing either its structure (Kirk-
+patrick et al., 2017; Ritter et al., 2018) or its response (Li
+and Hoiem, 2017; Schwarz et al., 2018); rehearsal meth-
+ods derive from the simple Experience Replay (ER) base-
+line, which exploits a working memory buffer to stash en-
+countered data-points, and later replays them when they
+are no longer available on the input stream (Robins, 1995;
+Chaudhry et al., 2019).
+Due to their versatility and effectiveness, current re-
+search efforts focus primarily on the latter class of ap-
+proaches (Aljundi et al., 2019). Recent trends highlight in-
+terest in improving several aspects of the basic ER formula,
+e.g., by introducing better-designed memory sampling
+strategies (Aljundi et al., 2019; Bang et al., 2021), combin-
+ing replay with other optimization techniques (Lopez-Paz
+and Ranzato, 2017; Riemer et al., 2019; Chaudhry et al.,
+2021) or providing richer replay signals (Buzzega et al.,
+2020a; Ebrahimi et al., 2021).
+One of the most prominent challenges for the enhancement
+of rehearsal methods is the imbalance between stream and
+replay data. Due to the reduced amount and variety of the
+latter, a continually learned classifier struggles to produce
+unified predictions and is instead biased towards the last
+learned classes (Hou et al., 2019; Wu et al., 2019).
+To
+counter this effect, researchers have come up with architec-
+tural modifications of the model (Hou et al., 2019; Douil-
+lard et al., 2020), purposed alterations to the learning ob-
+jective of the final classifier (Ahn et al., 2021; Caccia et al.,
+2022) or the outright removal of it, by applying representa-
+tion learning instead (Cha et al., 2021; Pham et al., 2021).
+Our proposal also aims at reducing the intrinsic bias of re-
+hearsal methods, but does so by enforcing a desirable prop-
+erty on the latent space of the model. This is achieved
+through a geometrically motivated regularization term that
+can be easily combined with any existing replay method.
+2.2
+Spectral geometry
+Our proposal is built upon the eigendecomposition of the
+Laplace operator on a graph, thus falling within the broader
+area of spectral graph theory. In particular, ours can be
+
+Frascaroli, Benaglia, Boschini, Moschella, Fiorini, Rodol`a, Calderara
+τ2 τ3 τ4 τ5 τ6 τ7 τ8 τ9 τ10
+Task
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+Label-Signal Variation (σ)
+iCaRL
+ER-ACE
+X-DER
++ CaSpeR
++ CaSpeR
++ CaSpeR
+X-DER
+X-DER + CaSpeR
+Task: τ6
+τ7
+τ8
+τ9
+(a)
+(b)
+Figure 2: Illustrations of how CL alters a model’s latent space. (a) A quantitative evaluation measured as Label-Signal
+Variation (σ) within the LGG for buffer data points – lower is better; (b) TSNE embedding of the features computed by
+X-DER for buffered examples in later tasks (top). Interference between classes is visibly reduced if CaSpeR is applied
+(bottom). All experiments are carried out on Split CIFAR-100, (a) uses buffer size 500, (b) uses 2000 (best seen in colors).
+regarded as an inverse spectral technique, as we prescribe
+the general behavior of some eigenvalues and seek a graph
+whose Laplacian spectrum matches this behavior.
+In the geometry processing area, such approaches take
+the name of isospectralization techniques and have been
+recently used in diverse applications such as deformable
+3D shape matching (Cosmo et al., 2019), shape explo-
+ration and reconstruction (Marin et al., 2020), shape mod-
+eling (Moschella et al., 2022) and adversarial attacks on
+shapes (Rampini et al., 2021). Differently from these ap-
+proaches, we work on a single graph (as opposed to pairs
+of 3D meshes) and our formulation does not take an input
+spectrum as a target to be matched precisely. Instead, we
+pose a weaker requirement: the gap between nearby eigen-
+values must be maximized, regardless of its exact value.
+Since our graph represents a discretization of the latent
+space of a CL model, this simple regularization has im-
+portant consequences on its learning process.
+3
+METHOD
+Our approach exploits tools from spectral geometry to reg-
+ularize the model’s latent space to hinder forgetting. In
+Sec. 3.1 we describe the Continual Learning paradigm; in
+Sec. 3.2 we present a preliminary experiment, highlighting
+the problem we want to address; finally, in Sec. 3.3 we il-
+lustrate our geometric regularizer.
+3.1
+Continual Learning Setting
+Following the CL criterion, the model Fθ is exposed incre-
+mentally to a stream of tasks τi, where i ∈ {1, 2, ..., T}.
+The parameters θ include both the weights of the feature
+extractor and the classifier, θf and θc respectively. Each
+task consists of a sequence of images and their correspond-
+ing labels τi = {(xi
+1, yi
+1), (xi
+2, yi
+2), ..., (xi
+n, yi
+n)} and does
+not contain data belonging to classes already seen in previ-
+ous tasks, so Y i ∩ Y j = Ø, with i ̸= j and Y i = {yi
+k}n
+k=1.
+At each step i, the model cannot freely access data from
+previous tasks and is optimized by minimizing a loss func-
+tion over the current set of examples:
+θ(i) = argmin
+θ
+ℓstream = argmin
+θ
+n
+�
+j=1
+ℓ
+�
+Fθ(xi
+j), yi
+j
+�
+,
+(1)
+where the parameters are initialized with the ones obtained
+after training on the previous task θ(i−1). If the model
+does not include mechanisms to prevent forgetting, the ac-
+curacy on all previous tasks will collapse. Rehearsal-based
+CL methods preserve a portion of examples from previous
+tasks and store them in a buffer B, with fixed size m. This
+data is then used by the model in conjunction with a spe-
+cific loss function ℓb to hamper catastrophic forgetting:
+θ(i) = argmin
+θ
+ℓstream + ℓb.
+(2)
+For instance, Experience Replay (ER) simply employs a
+cross-entropy loss over a batch of examples from B:
+ℓer ≜ CrossEntropy
+�
+Fθ(xb), yb�
+.
+(3)
+There exist different strategies for sampling the task data-
+points to fill the buffer. These will be explained in Sec. 4,
+along with detail on the ℓb employed by each baseline.
+3.2
+Analysis of changing Latent Space Geometry
+We are particularly interested in how the latent space
+changes in response to the introduction of a novel task
+on the input stream.
+For this reason, we compute the
+graph G over the latent-space projection of the replay ex-
+amples gathered by the CL model after training on τi
+(i ∈ {2, ...T})1. In order to measure the sparsity of the
+1Please refer to Sec. 3.3 for a detailed description of this pro-
+cedure.
+
+CaSpeR: Latent Spectral Regularization for Continual Learning
+Algorithm 1 CaSpeR Loss Computation
+Input Memory buffer B of saved samples
+1: xb ← BalancedSampling(B)
+2: zb ← Fθf (xb)
+3: A ← k-NN(zb)
+4: D ← diag(�b
+i a1,i, �b
+i a2,i, ..., �b
+i ab,i)
+5: L ← I − D−1/2AD−1/2
+▷ Eq. 5
+6: λ ← Eigenvalues(L)
+7: ℓCaSpeR ← −λg+1 + �g
+j=1 λj
+▷ Eq. 6
+Output ℓCaSpeR
+latent space w.r.t. classes representations, we compute the
+Label-Signal Variation σ (Lassance et al., 2021) on the ad-
+jacency matrix A ∈ Rm×m of G:
+σ ≜
+m
+�
+i=1
+m
+�
+j=1
+1yb
+i =yb
+jai,j ,
+(4)
+where 1· is the indicator function. In Fig. 2a, we evaluate
+several SOTA rehearsal CL methods and show they exhibit
+a steadily growing σ, which indicates that examples from
+distinct classes are increasingly entangled in later tasks.
+This effect can also be observed qualitatively by consider-
+ing a TSNE embedding of the points in B (shown in Fig. 2b
+for X-DER), which suggests that the distances between ex-
+amples from different classes are reduced in later tasks. We
+remark that both evaluations improve when our proposed
+regularizer is applied on top of the evaluated methods.
+3.3
+CaSpeR: Continual Spectral Regularizer
+Motivation. Our method builds upon the fact that the la-
+tent spaces of neural models bear a structure informative of
+the data space they are trained on (Shao et al., 2018). This
+structure can be enforced through loss regularizers; e.g.,
+in (Cosmo et al., 2020), a minimum-distortion criterion is
+applied on the latent space of a VAE for a shape genera-
+tion task. We follow a similar line of thought and propose
+adopting a geometric (namely, spectral-geometric) term to
+regularize the latent representations of a CL model.
+Our regularizer is based on the graph-theoretic formulation
+of clustering, where we seek to partition the vertices of G
+into well-separated subgraphs with high internal connec-
+tivity. A body of results from spectral graph theory, dating
+back at least to (Cheeger, 1969; Sinclair and Jerrum, 1989;
+Shi and Malik, 2000), explain the gap occurring between
+neighboring Laplacian eigenvalues as a quantitative mea-
+sure of graph partitioning. Our proposal, called Continual
+Spectral Regularizer (CaSpeR), draws on these results, but
+turns the forward problem of computing the optimal parti-
+tioning of a given graph, into the inverse problem of seek-
+ing a graph with the desired partitioning.
+Building the LGG. We take the examples in B and for-
+ward them through the network; their features are used to
+build a k-NN graph G; following (Lassance et al., 2021),
+we refer to it as the latent geometry graph (LGG).
+Spectral Regularizer. Let us denote by A the adjacency
+matrix of G, we calculate its degree matrix D and we com-
+pute its normalized Laplacian as:
+L = I − D−1/2AD−1/2 ,
+(5)
+where I is the identity matrix. Finally, we compute the
+eigenvalues λ of L and sort them by ascending order. Let
+g be the number of different classes within the buffer, we
+calculate our regularizing loss as:
+ℓCaSpeR ≜ −λg+1 +
+g
+�
+j=1
+λj .
+(6)
+The proposed loss term is weighted through the hyperpa-
+rameter ρ and added to the stream classification loss. Over-
+all, our model optimizes the following objective:
+argmin
+θ
+ℓstream + ℓb + ρ ℓCaSpeR .
+(7)
+Through Eq. 6, we increase the eigengap λg+1 − λg while
+minimizing the first g eigenvalues – the intuition being
+that the number of eigenvalues close to zero corresponds
+to the number of loosely connected partitions within the
+graph (Lee et al., 2014). Therefore, our loss indirectly en-
+courages the points in the buffer to be clustered without
+strict supervision. We refer the reader to Algorithm 1 for a
+step-by-step summary of the outlined procedure.
+Efficient Batch Operation. While seemingly straightfor-
+ward, the operation of CaSpeR entails the cumbersome task
+of constructing the entire LGG G at each forward step. In-
+deed, accurately mapping the model’s ever-changing latent
+space requires processing all available replay examples in
+the buffer B, which is typically orders of magnitude larger
+than a batch of examples on the input stream.
+To avoid a slow training procedure with high memory
+requirements, we propose an efficient approximation of
+our initial objective. Instead of operating on G directly,
+we sample a randomly chosen sub-graph Gp ⊂ G span-
+ning only p out of the g classes represented in the mem-
+ory buffer. As Gp still includes a conspicuous amount of
+nodes, we resort to an additional sub-sampling and extract
+Gt
+p ⊂ Gp, a smaller graph with t exemplars for each class.
+By repeating these random samplings in each forward step,
+we optimize a Monte Carlo approximation of Eq. 6:
+ℓ∗
+CaSpeR ≜
+E
+Gp⊂G
+�
+E
+Gtp ⊂Gp
+�
+− λ
+Gt
+p
+p+1 +
+p
+�
+j=1
+λ
+Gt
+p
+j
+��
+,
+(8)
+where the λGt
+p denote the eigenvalues of the Laplacian of
+Gt
+p. It must be noted that enforce the eigengap at p, as we
+know by construction that each Gt
+p comprises samples from
+p communities within G.
+
+Frascaroli, Benaglia, Boschini, Moschella, Fiorini, Rodol`a, Calderara
+4
+EVALUATION
+4.1
+Evaluation protocol
+Settings. To assess the effectiveness of the proposed
+method, we consider both split incremental classification
+protocols formalized in (van de Ven and Tolias, 2018):
+Task-Incremental Learning (Task-IL), where the task infor-
+mation is given during training and evaluation; and Class
+Incremental Learning (Class-IL), where the model learns
+to make predictions in the absence of task information. On
+the one hand, Class-IL is recognized as a more realistic and
+challenging benchmark (Farquhar and Gal, 2018; Aljundi
+et al., 2019); on the other, Task-IL is especially relevant for
+the quantification of forgetting, as it is unaffected by data
+imbalance biases (Wu et al., 2019; Boschini et al., 2022a).
+Benchmarked models. To evaluate the benefit of our
+regularizer, we apply it to the following state-of-the-art
+rehearsal-based methods:
+• Experience
+Replay
+with
+Asymmetric
+Cross-
+Entropy (ER-ACE) (Caccia et al., 2022): starting
+from classic Experience Replay, the authors obtain a
+significant performance gain by freezing the previous
+task heads of the classifier while computing the loss
+on the streaming data;
+• Incremental Classifier and Representation Learn-
+ing (iCaRL) (Rebuffi et al., 2017):
+this method
+seeks to learn the best representation of data that
+fits a nearest-neighbor classifier w.r.t. class prototypes
+stored in the buffer;
+• Dark Experience Replay (DER++) (Buzzega et al.,
+2020a): another variant of ER, which combines the
+standard classification replay with a distillation loss;
+• eXtended-DER (X-DER) (Boschini et al., 2022a):
+a method which improves DER++ by addressing its
+shortcomings and focusing on organically accommo-
+dating future knowledge2;
+• Pooled
+Outputs
+Distillation
+Network
+(POD-
+Net) (Douillard et al., 2020):
+the authors extend
+iCarl’s classification method:
+their model learns
+multiple representations for each class and adopts two
+additional distillation losses.
+We remark that these approaches adopt different strate-
+gies for the construction of their memory buffer: X-DER,
+iCaRL and PODNet use a class-balanced offline sam-
+pling strategy; ER-ACE and DER++ use reservoir sam-
+pling (Vitter, 1985), which might lead to uneven class rep-
+resentation within the stored examples. Since CaSpeR re-
+lies on the availability of a minimum amount of samples
+2Specifically, we use the more effective baseline based on a
+Regular Polytope Classifier (Pernici et al., 2021)
+per class, we adjust the latter sampling strategy to enforce
+equity, as done in (Buzzega et al., 2020b).
+To have a better understanding of the results, we include the
+performance of the upper bound (Joint), obtained by train-
+ing on all classes together in a standard offline manner, and
+the lower bound (Finetune) obtained by training on each
+task sequentially without any method to prevent forgetting.
+Datasets. We conduct all the experiments on two com-
+monly used image datasets, splitting the classes from the
+main dataset into separate disjoint sets used to sequentially
+train the evaluated models.
+• Split CIFAR-100: CIFAR100 (Krizhevsky et al.,
+2009) contains 100 classes with 500 images per class,
+where each image has a dimension of 32 × 32. We
+split the dataset into 10 subsets of 10 classes each;
+• Split miniImageNet: miniImageNet (Vinyals et al.,
+2016) is a subset of the ImageNet dataset where each
+image is resized to 84 × 84. We use the 20 tasks per 5
+classes protocol.
+Metrics. We mainly quantify the performance of the com-
+pared models in terms of Final Average Accuracy ( ¯AF ),
+i.e., the average classification accuracy of the model at the
+end of the overall training process:
+¯AF ≜ 1
+T
+T
+�
+i=1
+aT
+i ,
+(9)
+where aj
+i is the accuracy of the model at the end of task
+j calculated on the test set of task τi and reported in per-
+centage value. To quantify the severity of the performance
+degradation that occurs as a result of catastrophic forget-
+ting, we propose a novel measure called Final Average Ad-
+justed Forgetting ( ¯F ∗
+F ), which we define as follows:
+¯F ∗
+F ≜
+1
+T − 1
+T −1
+�
+i=1
+a∗
+i − aT
+i
+a∗
+i
+,
+where a∗
+i =
+max
+t∈{i,...,T −1} at
+i, ∀i ∈ {1, . . . , T − 1}.
+(10)
+¯F ∗
+F is typically bounded in [0, 100]3, where the upper
+bound is given by a method that retains no accuracy on pre-
+vious tasks (as is the case for the Finetune baseline). This
+measure derives from the widely employed Forgetting met-
+ric (Chaudhry et al., 2018), which tends to be more forgiv-
+ing of those methods that do not properly learn the current
+task (Buzzega et al., 2020a).
+Hyperparameter selection. To ensure a fair evaluation,
+we train all the models with the same batch size and the
+3 ¯F ∗
+F might assume a negative value if the learner improves its
+accuracy on past tasks; this generally indicates a pathological case
+where the model did not fully exploit the input stream of data.
+
+CaSpeR: Latent Spectral Regularization for Continual Learning
+Table 1: Class-IL results – ¯AF ( ¯F ∗
+F ) – for SOTA rehearsal CL methods, with and without CaSpeR.
+Class-IL
+Split CIFAR-100
+Split miniImageNet
+Joint (UB)
+63.11±2.07 (−)
+52.76±1.10 (−)
+Finetune (LB)
+8.38 (100.00)
+3.87 (100.00)
+Buffer Size
+500
+2000
+2000
+5000
+ER-ACE
+35.63 (45.03)
+46.63 (28.78)
+20.31 (39.06)
+26.17 (28.99)
++ CaSpeR
+36.70+1.07 (46.61)
+47.74+1.11 (27.17)
+23.36+3.05 (47.90)
+27.89+1.72 (28.36)
+iCaRL
+39.94 (32.24)
+40.95 (30.18)
+19.69 (36.89)
+20.78 (30.74)
++ CaSpeR
+40.50+0.56 (32.38)
+41.77+0.82 (28.81)
+20.31+0.62 (36.26)
+21.45+0.67 (37.26)
+DER++
+26.34 (66.13)
+45.68 (33.06)
+21.23 (71.76)
+28.94 (58.00)
++ CaSpeR
+31.66+5.32 (52.29)
+46.34+0.66 (30.08)
+21.48+0.25 (73.56)
+29.17+0.23 (57.69)
+X-DER
+35.89 (44.54)
+46.37 (23.57)
+24.80 (44.69)
+31.00 (30.12)
++ CaSpeR
+38.23+2.34 (43.90)
+50.39+4.02 (17.65)
+25.73+0.93 (42.93)
+31.39+0.39 (28.71)
+PODNet
+29.61 (55.06)
+32.12 (46.73)
+16.82 (52.32)
+20.81 (46.50)
++ CaSpeR
+31.29+1.68 (50.02)
+34.51+2.39 (40.66)
+17.14+0.32 (50.33)
+21.78+0.97 (46.74)
+same number of epochs. Moreover, we employ the same
+backbone for all experiments on the same dataset.
+In
+particular, we use Resnet18 (He et al., 2016) for Split
+CIFAR-100 and EfficientNet-B2 (Tan and Le, 2019) for
+Split miniImageNet. The best hyperparameters for each
+model-dataset configuration are found via grid search.
+We refer the reader to the Appendix for additional details.
+4.2
+Experimental results
+We report a breakdown of the results of our evaluation in
+Tab. 1 (Class-IL) and 2 (Task-IL). At first glance, CaSpeR
+leads to a steady improvement in ¯AF across all evaluated
+methods and settings.
+However, some interesting addi-
+tional trends emerge upon closer examination.
+Firstly, we notice that the improvement in accuracy does
+not grow with the memory buffer size.
+This is in con-
+trast with the typical behaviour of replay regularization
+terms (Cha et al., 2021; Chaudhry et al., 2019). We believe
+such a tendency to be the result of our distinctively geo-
+metric approach: as spectral properties of graphs are un-
+derstood to be robust w.r.t. to coarsening (Jin et al., 2020),
+CaSpeR does not need a large pool of data to be effective.
+Remarkably, the majority of the evaluated methods achieve
+comparable ¯AF gains for both CL settings on Split CIFAR-
+100; this suggests that our method allows the model to bet-
+ter learn and consolidate each task individually (Task-IL)
+while providing balanced responses for both stream and
+replay classes (Class-IL). This second tendency is further
+confirmed by the conspicuous reduction in Class-IL ¯F ∗
+F ,
+which confirms that CaSpeR counteracts the learning bias,
+whereby the learner predominantly focuses on the classes
+on the input stream.
+While still improving over the baselines, we see a reduced
+¯AF improvement in the Split miniImageNet benchmark.
+The mixed ¯F ∗
+F results in Class-IL might suggest that our
+approach is not particularly beneficial when it comes to
+comparing classes learned at different tasks. We suspect
+this might be a byproduct of our approximated batch op-
+eration, which only considers a few classes at any given
+training step and therefore struggles when dealing with the
+increased amount of tasks in this dataset.
+Even so, the
+Task-IL values for ¯F ∗
+F are favorably reduced, meaning that
+CaSpeR lets the model learn individual tasks more accu-
+rately so that it aptly recovers its predictive capability when
+cued with the correct task.
+As a final note, PODNet appears to be an outlier; with
+lower ¯AF and higher ¯F ∗
+F with respect to the other evalu-
+ated approaches, it exhibits a marked tendency to overfit
+current training data. Nevertheless, CaSpeR is still capable
+of impacting its training positively, delivering a stabilizing
+effect that is especially relevant when the memory buffer
+is large. This suggests that the additional clustering facil-
+itates the model’s convergence, which aligns with the ob-
+servations we make in Sec. 5.3, where we exploit CaSpeR
+with limited supervision.
+5
+MODEL ANALYSIS
+5.1
+k-NN classification
+To further verify whether CaSpeR successfully separates
+the latent embeddings for examples of different classes, we
+evaluate the accuracy of k-NN-classifiers (Wu et al., 2018)
+trained on top of the latent representations produced by the
+methods of Sec. 4. In Tab. 3, we report the results for 5-NN
+and 11-NN classifiers using the final buffer B as a support
+set. We observe that CaSpeR also shows its steady bene-
+ficial effect on top of this classification approach, further
+confirming that it is instrumental in disentangling the rep-
+resentations of different classes.
+
+Frascaroli, Benaglia, Boschini, Moschella, Fiorini, Rodol`a, Calderara
+Table 2: Task-IL results – ¯AF ( ¯F ∗
+F ) – for SOTA rehearsal CL methods, with and without CaSpeR.
+Task-IL
+Split CIFAR-100
+Split miniImageNet
+Joint (UB)
+88.81±0.84 (−)
+87.39±0.46 (−)
+Finetune (LB)
+30.10 (62.84)
+24.05 (67.37)
+Buffer Size
+500
+2000
+2000
+5000
+ER-ACE
+73.86 (10.73)
+80.69 (4.02)
+69.34 (12.99)
+73.38 (8.59)
++ CaSpeR
+75.14+1.28 (4.91)
+81.51+0.82 (4.38)
+69.59+0.25 (13.05)
+73.41+0.03 (8.53)
+iCaRL
+78.38 (5.38)
+78.47 (3.98)
+70.35 (3.92)
+70.99 (2.82)
++ CaSpeR
+79.09+0.71 (4.46)
+79.43+0.96 (3.41)
+71.19+0.84 (3.67)
+71.93+0.94 (3.65)
+DER++
+68.55 (12.24)
+79.60 (3.96)
+69.15 (13.22)
+73.81 (8.59)
++ CaSpeR
+72.40+3.85 (9.28)
+80.78+1.18 (3.04)
+70.07+0.92 (12.47)
+74.32+0.51 (7.91)
+X-DER
+77.28 (2.43)
+82.55 (0.92)
+74.32 (4.95)
+77.70 (3.71)
++ CaSpeR
+78.26+0.98 (5.47)
+83.77+1.22 (0.27)
+75.99+1.67 (3.88)
+78.71+1.01 (2.32)
+PODNet
+68.37 (18.76)
+67.63 (18.16)
+59.60 (14.00)
+64.15 (10.71)
++ CaSpeR
+69.07+0.70 (18.85)
+71.90+4.27 (11.32)
+60.06+0.46 (10.61)
+69.24+5.09 (8.18)
+ODE.85
+ER-ACE
+ODE.75
+ER-ACE + CaSpeR
+ODE.83
+X-DER
+ODE.64
+X-DER +CaSpeR
+ODE.81
+iCaRL
+ODE.79
+iCaRL + CaSpeR
+0.
+.2
+.4
+.6
+.8
+1.
+Fn. Map Magnitude (C|·|)
+Figure 3: For several rehearsal methods with and without CaSpeR, the functional map magnitude matrices C|·| between
+the LGGs G5 and G10, computed on the test set of τ1, ..., τ5 after training up to τ5 and τ10 respectively (Split CIFAR-100
+- buffer size 2000). The closer C|·| to the diagonal, the less geometric distortion between G5 and G10. We report the first
+25 rows and columns of C|·|, focusing on smooth (low-frequency) correspondences (Ovsjanikov et al., 2012), and apply a
+C|·| > 0.15 threshold to increase clarity.
+Table 3: Class-IL ¯AF values of k-NN classifiers trained
+on top of the latent representations of replay data points.
+Results on Split CIFAR-100 for Buffer Size 2000.
+k-NN Clsf
+w/o CaSpeR
+w/ CaSpeR
+(Class-IL)
+5-NN
+11-NN
+5-NN
+11-NN
+ER-ACE
+43.73
+44.41
+46.75+3.02
+47.29+2.88
+iCaRL
+34.86
+37.78
+36.00+1.14
+38.33+0.55
+DER++
+44.21
+44.24
+45.75+1.54
+46.00+1.76
+X-DER
+43.44
+44.62
+49.47+6.03
+49.49+4.87
+PODNet
+21.11
+22.60
+27.88+6.77
+28.94+6.34
+5.2
+Latent Space Consistency
+To provide further insights into the dynamics of the latent
+space on the evaluated models, we study the emergence of
+distortions in the LGG. Given a continual learning model,
+we are interested in a comparison between G5 and G10, the
+LGGs produced after training on τ5 and τ10 respectively,
+computed on the test set of tasks τ1, ..., τ5.
+The comparison between G5 and G10 can be better under-
+stood in terms of the node-to-node bijection T : G5 →
+G10, which can be represented as a functional map matrix
+C (Ovsjanikov et al., 2012) with elements
+ci,j ≜ ⟨φG5
+i , φG10
+j
+◦ T⟩ ,
+(11)
+where φG5
+i
+is the i-th Laplacian eigenvector of G5 (simi-
+larly for G10), and ◦ denotes the standard function compo-
+sition. In other words, the matrix C encodes the similarity
+between the Laplacian eigenspaces of the two graphs. In
+an ideal scenario where the latent space is subject to no
+modification between τ5 and τ10 w.r.t. previously learned
+classes, T is an isomorphism and C is a diagonal ma-
+trix (Ovsjanikov et al., 2012). In a practical scenario, T
+is only approximately isomorphic and, the better the ap-
+proximation, the more C is sparse and funnel-shaped.
+In Fig. 3, we report C|·| ≜ abs(C) for ER-ACE, DER++,
+iCaRL and X-DER on Split CIFAR-100, both with and
+without CaSpeR. It can be observed that the methods that
+benefit the most from our proposal (ER-ACE, X-DER) dis-
+play a tighter functional map matrix. This indicates that
+
+CaSpeR: Latent Spectral Regularization for Continual Learning
+the partitioning behavior promoted by CaSpeR leads to re-
+duced interference, as the portion of the LGG that refers
+to previously learned classes remains geometrically con-
+sistent in later tasks. On the other hand, in line with the
+considerations made in Sec. 4.2, the improvement is only
+marginal for iCaRL. Its different training regime, which is
+less discriminative in nature, seemingly induces a limited
+amount of change on the structure of the latent space.
+To quantify the similarity of each C|·| matrix to the iden-
+tity, we also report its off-diagonal energy, computed as fol-
+lows (Rodol`a et al., 2017):
+ODE ≜
+1
+||C||2
+F
+�
+i
+�
+j̸=i
+c2
+i,j,
+(12)
+where || · ||F indicates the Frobenius norm. CaSpeR pro-
+duces a clear decrease in ODE, signifying an increase in
+the diagonality of the functional matrices.
+5.3
+Continual Semi-supervised Learning
+In Sec. 4.2, we shed light on some interesting properties
+of CaSpeR, i.e., its ability to operate well in a low-data
+regime and its role in facilitating the convergence of under-
+performing baselines. Both issues naturally emerge in the
+Continual Semi-Supervised Learning (CSSL) setting (Bos-
+chini et al., 2022b), a recently-proposed CL experimental
+benchmark, where only a fraction of the examples on the
+input stream are associated with an annotation.
+In a supervised CL setting, we apply CaSpeR to buffer data
+points, thus encouraging the separation of all previously en-
+countered classes in the latent space. However, we remark
+that our proposed approach does not have strict supervision
+requirements, as it does not need the labels attached to each
+node in the LGG, but rather just the total amount of classes
+g that must be clustered (Eq. 6).
+In Tab. 4, we report the results of an experiment on Split
+CIFAR-100 in the CSSL setting with only 0.8% or 5%
+annotated labels.
+Typical CL methods operating in this
+scenario are forced to discard a consistent amount of data
+(ER-ACE), leading to majorly reduced performance w.r.t.
+the fully-supervised case, or to use the in-training model to
+annotate unlabeled samples (pseudo-labeling, PsER-ACE),
+but might backfire if the provided supervision does not suf-
+fice for the learner to produce reliable responses (as is the
+case with 0.8% labels).
+To allow for the exploitation of unlabeled exemplars, we
+also apply CaSpeR on data points from the input stream,
+by taking k equal to the number of classes in a given task.
+We show that this leads to an overall improvement of the
+tested models and – particularly – counteracts the failure
+case where PseudoER-ACE is applied on top of a few an-
+notated data. This indicates that CaSpeR manages to limit
+the impact of the noisy labels produced by pseudo-labeling.
+Table 4: Class-IL ¯AF values on Split CIFAR-100, with re-
+duced amount of annotations (CSSL). Buffer size 2000. †
+indicates results taken from (Boschini et al., 2022b).
+CSSL
+w/o CaSpeR
+w/ CaSpeR
+Labels %
+0.8%
+5%
+0.8%
+5%
+ER-ACE
+8.46
+11.87
+8.55+0.09
+14.16+2.29
+PsER-ACE
+2.31
+16.35
+9.69+7.38
+17.42+1.07
+CCIC
+11.5†
+19.5†
+12.22+0.72
+20.32+0.82
+Finally, we show that CaSpeR can be easily applied to
+CCIC (Boschini et al., 2022b) – a CSSL method that lever-
+ages both labeled and unlabeled data – to improve its ¯AF .
+6
+CONCLUSION
+In this work, we investigate how the latent space of a CL
+model changes throughout training. We find that latent-
+space projections of past exemplars are relentlessly drawn
+closer together, possibly interfering and paving the way for
+catastrophic forgetting.
+Drawing on spectral graph theory, we propose Continual
+Spectral Regularizer (CaSpeR): a regularizer that encour-
+ages the clustering of data points in the latent space. We
+show that our approach can be easily combined with any
+rehearsal-based CL approach, improving the performance
+of SOTA methods on standard benchmarks.
+Furthermore, we analyze the effects of CaSpeR showing
+that the regularized latent space correctly separates exam-
+ples from different classes and is subject to fewer distor-
+tions. Finally, we verify that our proposed approach is also
+applicable with partial supervision, improving the accuracy
+of Continual Semi-Supervised Learning baselines and fa-
+cilitating their convergence in a low-label regime.
+Limitations & Societal Impact
+While our proposed regularizer can moderately operate
+without full supervision, we remark that it still depends on
+the availability of supervised training signals. The appli-
+cability of geometric-based constraints to unsupervised or
+self-supervised CL scenarios is still a work in progress.
+Due to the abstract nature of our setting, we do not believe
+that this work can have a detrimental impact on society.
+However, given the necessity for the proposed regularizer
+to store and re-use previously learned training samples, we
+remark that its applicability might be limited if privacy con-
+straints are in place.
+
+Frascaroli, Benaglia, Boschini, Moschella, Fiorini, Rodol`a, Calderara
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